The README of the skaters package opens with one sentence: “Laplace beats (almost) everything.” It adds that Laplace gives ground to my method, Conformal Seasonal Pools (CSP), only “narrowly”, and only because CSP was handed the seasonal period.

That headline misrepresents what was tested. The evidence behind it is one-step-ahead forecasts of series cherry-picked by Laplace’s author: economic change series from FRED and, in the comparison with CSP, one hourly collection besides. FRED holds thousands of series, but FRED is one dataset: one source, one kind of data. Counted the way every benchmark counts, the evidence is two datasets, one horizon and one family of scores. This study uses 228 datasets.

FRED is also about as narrow as a dataset can be. It is economic and financial indicators, most of them from one country, published daily at the fastest. It has no hourly or sub-hourly data, no electricity load, no traffic, no retail demand, no web traffic, no sensor or weather series, no cloud workloads. Its series are scored as changes from one observation to the next, which removes level, trend and most of the seasonality before the forecaster sees them. A method tuned on that is tuned for one corner of forecasting, and that corner is where every one of Laplace’s headline wins comes from.

That is not how a forecasting method is evaluated. Best practice, established by the M competitions and followed by every serious benchmark since, is large-scale testing: hundreds of datasets from many domains and frequencies, several forecast horizons, a set of metrics fixed in advance, and a corpus the method’s author did not choose. A claim to beat almost everything needs that kind of evidence. Its author has not published that test.

When I ran Laplace on such a benchmark, 228 datasets at multi-step horizons, CSP beat it on 60% of datasets by CRPS and on 64% by interval score, with intervals 30% narrower, and ran about 7,700 times faster.

It also produces forecasts no practitioner would accept: electricity demand predicted to go negative within a day, and intervals fifteen orders of magnitude wider than the data.

This article sets out what the claim rests on, what I tested, what came out, and what the forecasts look like. All the data and code are public.

What the claim rests on

The comparison with CSP in the skaters repository reports Laplace beating CSP on 82 to 99% of series in three collections and losing on the fourth. Read the protocol and the scope is narrow in five ways.

• One step ahead only. Every forecast is for the next observation. Multi-step forecasting, the usual practical task, is not tested.
• A narrow choice of data. Three of the four collections are economic change series from one source, FRED. The fourth is M4-Hourly. The series were chosen by Laplace’s author.
• Two metrics. Only CRPS and log-likelihood are reported. There is no interval score, no interval width, and no coverage for CSP.
• An old CSP. The repository comparison uses csp-forecaster 0.1.4. The current version is 0.2.0. The author’s website now shows a November re-run with 0.2.0, which is still one step ahead; I come to it below.
• Not period-agnostic. The README footnote presents Laplace as having to work out the seasonal structure unaided. Its code has built-in seasonal candidates for periods 7, 12 and 24, and M4-Hourly has period 24.

A result obtained this way can support “Laplace does well one step ahead on FRED economic series”. It cannot support “beats (almost) everything”.

What I tested instead

I ran Laplace on the benchmark from the CSP paper, which was fixed before Laplace was ever run on it.

The run covers 2,534 rolling-origin forecast windows with 95% prediction intervals. Both methods see exactly the same history at every forecast origin.

What came out

CSP is the more accurate method on about six datasets in ten. Scores are averaged within each dataset, and a method wins a dataset when its average is lower.

All five differences are statistically significant in a paired test across datasets.

Quantile loss is the metric of Amazon’s GluonTS library and of the DeepNPTS paper, and the one reported alongside CRPS in the CSP paper: the pinball loss of the forecast quantiles at levels 0.1 to 0.9, scaled by the size of the series. It agrees with CRPS. CSP wins 59% of datasets on it, and on a typical dataset its quantile loss is 36% lower than Laplace’s.

Interval width. The skaters comparison does not report it. Laplace’s 95% intervals are 44% wider than CSP’s (median scaled width 9.40 against 6.54).

Coverage. Laplace’s average coverage is 0.956 and CSP’s is 0.898, so Laplace is closer to the 95% target on average. It gets there by over-covering: on 76 of 228 datasets its intervals contain 99% or more of outcomes. CSP errs the other way and is below 85% on 39 datasets. The interval score charges for both mistakes, and it favours CSP on 64% of datasets.

Horizon. At horizons up to 12 steps the two are level: CSP wins 31 of 60 datasets by CRPS. CSP’s advantage is at horizons of 13 to 96 steps. A one-step test cannot see this.

Point forecasts. Laplace’s point forecast is biased low and, on these windows, is on average less accurate than a plain seasonal-naive forecast.

Speed

Laplace took 14,664 seconds for the 2,534 forecasts. CSP took 1.9 seconds. That is about 7,700 times slower, and the gap widens as the horizon grows.

These times are with a warm-up of only 400 to 720 observations. Laplace’s cost per observation grows roughly with the square of the horizon, and at long horizons it becomes impractical to run. That is why this test covers 2,534 of the benchmark’s 24,502 windows, why histories were truncated, and why the dataset with a horizon of 336 was left out: by my cost estimate a single forecast there would take about six minutes. CSP runs the whole benchmark in about 40 seconds.

Six forecasts to look at

Win rates can hide what a forecast actually looks like. In 2021 Laplace’s author wrote a well-known critique of Facebook’s Prophet. Its method was simple: show real forecasts on real series, ask whether a sensible person would have drawn them, and check whether the model beats a naive baseline. The same three tests are applied to Laplace here.

The figure shows six forecasts from the benchmark. They were picked because they are failures. They are not typical, and the win rates above say how often Laplace does better than this. Each panel shows the history, what then happened, Laplace’s median forecast and 95% interval, and CSP’s 95% interval from the same history.

Six of Laplace’s worst forecasts on the benchmark: history, outcome, Laplace’s median and 95% interval, and CSP’s 95% interval

• Electricity demand forecast to go negative. On Australian half-hourly demand, the Texas grid (ERCOT West) and the Taylor load series, Laplace’s median forecast leaves the daily cycle, falls in a straight line and crosses zero within a day. On the Taylor series, where load runs between 21,000 and 37,000, the lower end of its 95% interval reaches –109,000.
• An interval ten times wider than the series. On the M4 hourly series the data move between 294 and 800. Laplace’s 95% interval runs from –4,242 to 1,763. Its median forecast misses by 874 on average; CSP’s misses by 26.
• Numbers with 35 digits. On the Bitcoin hash rate, which is around 1020, Laplace’s interval reaches 1035 and its median forecast jumps between plus and minus 1030 from one day to the next. On a GDP series around 1010 the interval reaches 1017.

The pattern in the top four panels is the same. Each history has an obvious daily cycle. At multi-step horizons Laplace does not carry it forward: it extends the last movement of the series and lets the interval widen with the horizon, as a random walk would.

Two simple counts show these are not isolated accidents.

• Negative forecasts of non-negative series. In 2,460 of the windows every outcome is zero or above. In 187 of them, about one in thirteen, Laplace’s median forecast goes below zero. CSP’s does so in 12.
• Losing to the naive forecast. On 112 of 227 datasets, 49%, Laplace’s point forecast is less accurate than simply repeating the last season. For CSP the figure is 75 datasets, 33%.

The public claims, one by one

The claims are made in the skaters repository, on the author’s websites and in his LinkedIn posts. I saved a copy of each on 12 and 13 November 2026. Here they are in turn, with what the data says. Claims are paraphrased unless in quotation marks.

1. “Laplace beats (almost) everything.” Source: skaters README. The evidence is one step ahead, on FRED change series and M4-Hourly. On 228 datasets at multi-step horizons Laplace loses to CSP, a baseline of a few lines of numpy, on 60% of datasets by CRPS and on 64% by interval score.

2. Laplace gives ground to CSP only narrowly, and only on repeating waveforms. Source: skaters README. In this test the gap is not narrow and is not confined to seasonal data. On a typical dataset CSP’s CRPS is 35% lower than Laplace’s. On the 45 datasets with no seasonal period CSP still wins 26 by CRPS.

3. That loss is a head start, because CSP is given the seasonal period and Laplace is not. Source: README footnote. Laplace ships with built-in seasonal candidates for periods 7, 12 and 24, which covers the hourly collection in question. And CSP wins most of the non-seasonal datasets, where knowing a period gives it nothing.

4. Laplace beats CSP on 82 to 99% of FRED series by CRPS. Source: the repository comparison; the website gives 77 to 99%. This is one slice of the forecasting problem: one-step forecasts of economic change series from a single source, cherry-picked by Laplace’s author. A result on a slice chosen by a method’s author is not evidence about how the method performs in general. That is settled on a large benchmark covering many domains, frequencies and horizons, fixed in advance. On such a benchmark, 228 datasets across ten domains, the result reverses.

5. A November re-run against CSP 0.2.0 confirms the pattern. Source: the author’s review of CSP on conformalprediction.net. The re-run is again one step ahead. Three of its four panels are FRED. More than half of its 13,287 series are daily prices, a class of series for which the author himself says Laplace is not recommended. On the fourth panel, M4 hourly, his own per-series test favours CSP on 144 series and Laplace on 19.

6. The interval score is the wrong test and datasets are the wrong unit. Source: the same page. The interval score is a proper score and was the interval metric of the M4 and M5 competitions. CRPS, which the page prefers, also favours CSP on 60% of datasets. The dataset is the right unit for a benchmark of this kind: series and windows from one dataset are not independent, and counting them one by one lets a few large collections decide the result.

The page objects to the interval score and offers CRPS in its place. CRPS has well-documented weaknesses of its own, and it is not safe to rank by it alone. It gives every quantile level the same weight, so errors in the tails, which is where a 95% interval lives, move it very little; this is why Gneiting and Ranjan proposed weighted versions in 2011. It is also measured in the units of the data, so an average is dominated by the largest series unless it is scaled.

The CSP paper ran into exactly this. DeepNPTS’s 95% intervals contained the outcome only 57% of the time on its own six benchmark datasets, yet on CRPS it ranked fourth of six methods, in the middle of the field. On the interval score it ranked at the bottom. Ranked by CRPS alone, a forecaster whose intervals fail almost half the time looks ordinary. That is why a serious comparison reports CRPS together with the interval score, coverage and width, and why this one does. A comparison that reports CRPS and log-likelihood only cannot show whether the intervals work.

Claims 5 and 6 were not on the page when it first appeared. The review of CSP was published on 12 November 2026 at 16:32 GMT and amended the same evening at 19:52 GMT. The amendment added the section arguing against the interval score and against counting datasets, added the November table, and rewrote the footnote. The site is built from a public repository, so both versions are on record: the original and the amendment.

7. Nobody has won a time-series contest with conformal prediction, “and nobody ever will”. Source: LinkedIn, November 2026. On this benchmark a conformal sampler with no training has the better CRPS than his own model on 137 of 228 datasets. False: the first ever probabilistic forecasting competition on Kaggle was won by conformal prediction. The winner Stephane Degeye has written about it in his article on Medium

8. Laplace is better than every foundation and traditional distributional model he can find, across 18,815 series, at all horizons up to 13. Source: LinkedIn, August 2026. The evidence stops at 13 steps. At horizons up to 12 my test also finds the two level. From 13 to 96 steps CSP wins 104 of 161 datasets by CRPS.

9. Laplace averages 12 microseconds per observation and is about 370,000 times faster than a leading foundation model. Source: LinkedIn, November and August 2026. The posts do not say so, but the author’s own performance page shows where 12 microseconds comes from: the Rust port, one step ahead, updating a model that is already warm. The same page gives the Python package 23 times that for one step, and 74 milliseconds to forecast a series it has not seen before. At the multi-step horizons of this benchmark the Python package took between 7 and 224 milliseconds per observation and 0.75 to 107 seconds per forecast. It was 2,500 to 23,800 times slower than CSP.

10. On 5,402 FRED series Laplace wins the log-likelihood race against classical forecasters, conformal methods and all zero-shot foundation models. Source: LinkedIn, August 2026. The metric is log-likelihood, one step ahead. His later posts withdraw the last part: in October he conceded that one foundation model is better on weekly and waveform data, and a footnote in November says two are ahead overall.

The pattern is the same throughout. The strong statements are general. The evidence is one step ahead, on economic series, scored mainly by log-likelihood. None of these sources reports interval width or an interval score for Laplace against a conformal method.

Where Laplace wins, and where it breaks

Laplace is not a bad method, and it wins 40% of the datasets. Its clearest wins are Jena weather, where its CRPS is about a tenth of CSP’s, FRED-MD, rideshare and M3 quarterly. Economic and slowly varying series suit it, which is what its own benchmark shows.

Laplace’s problems, in full

These are the weaknesses I measured, each on skaters 0.16.0 at default settings.

1. Multi-step forecasting is very slow, and gets slower with the horizon. The model has to be fed a series one point at a time, and its cost per observation grows roughly with the square of the horizon: about 7 milliseconds per observation at horizons up to 12, and about 224 beyond 96. One forecast takes 0.75 seconds at short horizons and 107 seconds at long ones, after a warm-up of only 400 observations.

2. It cannot be run at long horizons in reasonable time. At a horizon of 336 a single forecast would take about six minutes. A benchmark of 24,502 windows would need days of computing for Laplace. CSP needs about 40 seconds.

3. Its accuracy falls behind as the horizon grows. Up to 12 steps Laplace and CSP are level (CSP wins 31 of 60 datasets by CRPS). From 13 to 96 steps CSP wins 104 of 161, which is 65%.

4. It does not carry a seasonal cycle forward. On an M4-Hourly series at a horizon of 48, Laplace’s median forecast had a mean absolute error of 381, against 57 for the seasonal-naive forecast. Its 95% interval was 138 wide at step 1 and 5,526 wide at step 48, on a series that ranges from 412 to 899. Across the 184 seasonal datasets CSP wins 112 by CRPS and 125 by interval score.

5. Its forecast distribution blows up on some series. On Bitcoin the widest 95% interval was about 1035. On a global-economy panel it was about 1017, and on CIF 2016 about 1013. Laplace’s CRPS is more than ten times CSP’s on 9 datasets and more than twice on 37. CSP did not blow up on any window.

6. Its intervals are wider than they need to be. They are 44% wider than CSP’s at the median, and on 76 of 228 datasets they contain 99% or more of the outcomes at a nominal 95%. That is why Laplace loses the interval score on 64% of datasets.

7. Nothing bounds its coverage. Laplace is an ensemble with a calibration layer and has no coverage guarantee. Its average coverage is good (0.956), but it is below 0.85 on 13 datasets, and as low as 0.75 on M3 yearly and 0.78 on monthly sunspots.

8. Its point forecast is biased low. The median bias is –0.24 in units of the naive forecast’s error, against –0.02 for CSP. Its median MASE is 1.21, so on the typical dataset it is less accurate than the seasonal-naive forecast.

9. One step ahead, it under-covers on hourly data. In a separate one-step check on M4-Hourly its 90% interval covered 88.0%, with 18% of series below 85%. CSP covered 91.5%, with 1% of series below 85%.

10. On macroeconomic series its margin over a naive floor is small. In the same one-step check on FRED-MD, Laplace’s median CRPS was 5.5% lower than that of a last-value forecast with a rolling conformal interval, and it lost to that floor on 15% of series.

The seasonal anchor

In July 2026 the Laplace of the time lost to CSP’s default configuration on 91% of the M4-Hourly series by CRPS. That figure is from the skaters benchmark notes. Its author then added a seasonal_anchor component to Laplace, and his own page on CSP says Laplace “adopted a seasonal anchor as a result”.

The anchor is an equal blend of a recency-weighted average of same-phase values and the value one season ago. That is the structure of CSP’s default sampler: a recency-weighted same-phase pool mixed in equal parts with a component centred on the seasonal-naive forecast.

By the author’s own figures the anchor improved Laplace’s median CRPS on M4-Hourly by 8.3%. So Laplace lost to CSP, took the structure that beat it, and improved as a result. CSP is not credited for it. The README does not name CSP as the source. The documentation of the component in the code does not mention CSP at all.

The NNS attribution

Instead, the component’s documentation credits the NNS package, an R package, for forecasting from same-phase series, and gives the date as 2017. The benchmark notes go further and describe CSP’s seasonal pool as a draw from the decomposition found in NNS.

This attribution is wrong in three ways.

• The idea is far older than NNS. Forecasting each position in the cycle from its own history is what seasonal indices do in Holt-Winters exponential smoothing (1960), what periodic autoregressive models do (1967), and what the cycle-subseries step of STL does (1990). The seasonal-naive forecast is the simplest case of it. Presenting NNS as the source is out by more than half a century.
• The date is wrong too. NNS has had same-phase component series since its first CRAN release in April 2016, not 2017.
• NNS does not contain what Laplace adopted. NNS regresses each same-phase series on time, or takes its plain mean. It does not blend a recency-weighted same-phase value in equal parts with the seasonal-naive value. That blend is CSP’s default, and it is what the seasonal anchor is.

Nobody owns the same-phase idea, and CSP does not claim it. The specific construction Laplace adopted, after losing to CSP, is CSP’s. Citing a different package for it, and for an idea that package did not originate either, leaves the reader with a false picture of where the component came from.

What should change

1. Scope the headline. “Beats (almost) everything” should say what was tested: one step ahead, on FRED change series and M4-Hourly.
2. Report interval width, interval scores, coverage, runtime and multi-step results in the CSP comparison.
3. Cite CSP for the seasonal anchor.

Imagine a heap of coins. You count them out in threes and two are left over. In fives, three are left. In sevens, two. How many coins are in the heap?

The answer is 23. The question is far older than the answer key. It first appears in the Chinese mathematician Sun Zi, who lived around the third century AD. Today we call the general method behind it the Chinese Remainder Theorem, one of the workhorses of number theory and modern cryptography.

Gauss gave the theorem its complete proof in 1801. What almost nobody outside Russia knows is that more than a century earlier, Russian arithmeticians were already using the full method. The puzzle circulated in Muscovite manuscripts of the seventeenth century, under a wonderfully practical title: “Of knowing the money in a heap.” And in one of those manuscripts, a Russian compiler did something that, as far as historians have found, no Western European arithmetic book of the period did. He precomputed every answer and put them in a table.

A puzzle that crossed a continent

The problem’s route to Russia is a small history of the world in itself. A. P. Yushkevich traces it in History of Mathematics in Russia Before 1917:

• China, c. 3rd century AD. Sun Zi poses the 3–5–7 problem. Chinese mathematicians already possessed a general method.
• Byzantium. The problem turns up in a Greek manuscript of the late fourteenth or early fifteenth century.
• Italy, 1202. Leonardo of Pisa (Fibonacci) includes it in Liber Abaci, probably picked up on his travels in the Arab world and Byzantium.
• Germany and France. German arithmetics of the fifteenth century carry it. Bachet de Méziriac treats it in 1612.
• Euler works on it. Then, in 1801, Carl Friedrich Gauss gives a complete justification of the method for any number of divisors, in the language of congruences he invented.

Somewhere along the way, by paths no one has traced, it reached Moscow.

The Russian recipe: 70, 21, 15 and 105

Most Russian manuscripts state the rule as a bare recipe. Count the heap in threes: for every coin left over, write down 70. Count in fives: for every coin left over, write down 21. Count in sevens: for every coin left over, write down 15. Add everything up and keep subtracting 105 until you can’t. What remains is the heap.

Try it on Sun Zi’s heap, with remainders 2, 3 and 2:

2 × 70 + 3 × 21 + 2 × 15 = 140 + 63 + 30 = 233

233 − 105 − 105 = 23.

It works because each “magic” number does one job and stays out of the way of the others. 70 leaves remainder 1 when divided by 3, but is divisible by both 5 and 7. 21 leaves remainder 1 when divided by 5, but is divisible by 3 and 7. 15 leaves remainder 1 when divided by 7, but is divisible by 3 and 5. Multiply each by its remainder, add, and every condition is satisfied at once. The 105 is simply 3 × 5 × 7, the point where the pattern repeats.

In modern notation, that is the Chinese Remainder Theorem: if x ≡ r₁ (mod q₁), x ≡ r₂ (mod q₂) and x ≡ r₃ (mod q₃) for pairwise coprime divisors, then x is the remainder of N₁q₂q₃r₁ + N₂q₁q₃r₂ + N₃q₁q₂r₃ on division by q₁q₂q₃, where each N is chosen so that its term leaves remainder 1 for “its own” divisor.

A student’s notebook gives the game away

A bare recipe proves nothing about understanding. Anyone can memorise 70, 21, 15. Could seventeenth-century Russian arithmeticians actually derive those numbers for divisors they had never seen?

Yushkevich argues that they could, and the evidence comes from an unlikely source: a manuscript known as Counting Wisdom, which he identifies as a student’s work. The language is close to speech, the computations are unusually detailed, the abbreviations and slips are everywhere. It reads, he says, as if the writer were retelling what the teacher said in class.

And in it, the method is spelled out. The multipliers are called “examples.” The product of the divisors is the “dividing number.” To find the “example” for 3, multiply the other two divisors, 5 × 7 = 35, then try 35, 2 × 35, 3 × 35 and so on, dividing by 3 each time, until a remainder of 1 appears. That is exactly the search a modern textbook describes.

The notebook then runs the method on divisor triples well beyond Sun Zi’s:

I re-derived every one of these numbers. All of them are correct.

The table

The most striking artefact is in a manuscript held in the State Hermitage (shelfmark É/RB–40, folio 24). After the usual recipe comes a separate sheet carrying a three-way table, written in Slavonic numerals. It lists the smallest heap for every combination of remainders on division by 3, 5 and 7–105 numbers in all. A short instruction beneath explains how to read it: find the row for the remainder by 3 and by 5, then go along to the column for the remainder by 7.

Want the heap that leaves 2, 3 and 4? Read off 53.

The table has a hidden rhythm. Along each row the entries climb by 15. Down each column they climb by 91, wrapping round past 105. Whoever built it understood the structure, not just the recipe.

Yushkevich is careful here. Western European arithmetic books gave the same rule for the multipliers. But he found no solution laid out as a table in them, and concludes that the table was apparently the work of a Russian mathematician. In the language of modern engineering, it is a precomputed cache: do the arithmetic once, then answer any query by lookup.

Why would anyone need 15, 19 and 28?

One of those divisor triples is stranger than the rest. 15, 19 and 28 multiply to 7,980. They are not a natural classroom exercise.

The book does not explain the choice, so what follows is my reading, not Yushkevich’s. Those three numbers are the classic cycles of Christian calendar reckoning: the 15-year indiction, the 19-year lunar cycle and the 28-year solar cycle. Together they repeat every 7,980 years, the period Joseph Scaliger named the Julian Period in 1583. Given a year’s position in all three cycles, the Chinese Remainder Theorem recovers the year itself. Calendar computation was a living skill in a church culture that had to fix Easter. A scribe who needed it had every reason to copy the multipliers 6916, 4845 and 4200.

So — did Russians solve it before Gauss?

Honestly: yes and no.

No, they did not get there first. Chinese mathematicians had a general method well over a thousand years earlier, Fibonacci published the problem in 1202, and Euler worked on it before Gauss.

Yes, in a narrower and still remarkable sense. More than a century before Gauss’s Disquisitiones Arithmeticae (1801), Russian arithmeticians were using the complete algorithm for three divisors, deriving the multipliers for new cases, teaching it in class — and at least one of them turned it into a full lookup table that no Western arithmetic book of the time appears to contain.

Gauss gave the theorem its proof and its general form. The Russian scribes gave it something else: a working tool, built for people who needed answers.

Where this story comes from

This episode is one of hundreds in A. P. Yushkevich’s History of Mathematics in Russia Before 1917, the standard Soviet history of the subject, from birch-bark numerals in Novgorod to Chebyshev, Markov, Lyapunov, Kovalevskaya and the birth of the Moscow school. It has never been available in English until now.

The first faithful English translation from the original Russian edition is available here:
https://valeman.gumroad.com/l/history_of_mathematics_in_russia_before_1917

Source: A. P. Yushkevich, History of Mathematics in Russia Before 1917, Chapter 2, §“The problem ‘of knowing the money in a heap’” (pp. 37–40), translated by Valery Manokhin, Northern Star Academic Press. Yushkevich’s own detailed study is “On a problem of number theory in Russian mathematical manuscripts of the seventeenth century,” Trudy Inst. ist. estestv. i tekhn., vol. 17 (1957).

Leonhard Euler had an operation that was supposed to save his sight.

It worked. Briefly. Then, in the words of the man who would deliver his eulogy, “he neglected the necessary precautions, by rushing his recovery and lost his sight after terrible pain and suffering.”

He was the most productive mathematician who has ever lived, and he could no longer read a line of his own work.

The mathematics did not stop. That is the part people find hard to believe. Euler kept producing at a rate that would embarrass a modern department, working “by the sole intermediary of his memory and imagination.” What he needed was not a nurse. He needed hands, and a mind attached to them — somebody who could take dictation in mathematics, spot the slip before it reached the page, run the tables, build the examples.

He wrote to Basel, to his old friend Daniel Bernoulli.

Bernoulli knew exactly who to send.

The boy from Basel

Nicolas Fuss was born in Basel on 29 January 1755. By his teens his mathematical ability had already come to Bernoulli’s attention — enough that when Euler needed a secretary, Bernoulli put forward a local youth with no reputation, no publications, and no connections in Russia.

Fuss arrived in St Petersburg in May 1773. He was eighteen.

He never left. He spent the remaining fifty-three years of his life in Russia.

What an assistant actually did

Here is the number that reframes the job: Fuss helped Euler prepare more than 250 articles for publication.

Not copying. He “developed his ideas, calculating tables, and sometimes compiled examples.” When a blind man dictates a proof, somebody has to hold the whole structure in their head at the same time he does, or the errors compound silently and the work is lost.

Fuss did that, in his late teens and twenties, for the mind that produced the Basel problem, the seven bridges, e, and the identity that carries Euler’s name.

He also married into the family. His wife was Albertine Benedikte Philippine Luise Euler, Leonhard’s granddaughter.

The eulogy

On 23 October 1783, three weeks after Euler died, the Imperial Academy of Sciences of Saint Petersburg gathered to hear his eulogy.

It was delivered by Fuss.

It is a remarkable document — the definitive contemporary account of Euler’s life, written by the person who had sat closest to him for a decade. Fuss describes how the blind Euler worked, how “his prodigious and astonishing memory, enforced by the concentration of all of its forces disengaged from external distraction,” carried the work that his eyes no longer could.

And when he lists the people who lent Euler their hands in those final years, he names Euler’s sons, and Krafft, and Lexell.

He leaves himself out.

The assistant becomes the scientist

Euler died in 1783. Fuss was made an academician in higher mathematics the same year.

What followed was a full career, not a footnote to someone else’s. He worked in spherical geometry, trigonometry, series, differential geometry and differential equations. MacTutor rates his best work as the spherical trigonometry he did with Anders Johan Lexell and Friedrich Theodor Schubert. There is a theorem named after him — Fuss’s theorem, on bicentric quadrilaterals — and a sequence of numbers, the Fuss–Catalan numbers, that turns up in modern combinatorics and free probability.

From 1790 he was professor of mathematics at the Marine Corps in Petersburg. From 1800 until his death in 1826 he was permanent secretary of the Imperial Academy of Sciences — the administrative centre of Russian science for a quarter of a century.

His son Paul Heinrich Fuss joined the Academy in 1823, edited the first attempt at Euler’s collected works, and became the Academy’s secretary in 1826, the year his father died.

Father to son. Secretary to secretary. Euler’s papers still moving through the same family.

The thing almost nobody knows he did

Late in life, Fuss wrote schoolbooks.

Not a curiosity — a trilogy, for the schools of the Russian Empire, published by the state school authority. Part One: algebra, 1820. Part Two: geometry, 1823. Part Three: the application of algebra to geometry, plane trigonometry, conic sections, and the foundations of the differential and integral calculus.

The first volume says on its own title page what it is: the elementary foundations of algebra, extracted from the foundations of that science laid down by the celebrated Euler.

Read that again. The man who had been Euler’s eyes for a decade took Euler’s algebra and rebuilt it as a course for children.

Four divisions. Fifty-nine chapters. Five hundred and seventy numbered sections. It starts with what it means for a letter to stand for a quantity and ends at the irreducible case of the cubic and the approximate solution of equations of any degree — by way of logarithms, imaginary numbers, infinite series, the permutation proof of the binomial, and the full treatment of the fourth degree.

Why this book is the root

Russian school mathematics has a reputation. The Soviet mathematical school, the olympiad tradition, Kiselev’s textbooks that stayed in classrooms for seventy years — people know the fruit.

This is the root.

Fuss’s course is the channel through which Euler’s algebra reached Russian classrooms. Everything downstream — Kiselev above all — is built on the ground it laid. If you have ever wondered why the Russian tradition teaches algebra as a connected argument rather than a bag of procedures, the answer starts in 1820, with a man who learned his algebra by sitting next to Euler and writing it down.

Two hundred years

In all that time, the book has never been published in English. Not a partial translation. Not an abridgement. Nothing.

That ends now. Fuss’s Elementary Foundations of Pure Mathematics, Part I: Algebra is available in paperback and hardcover — the complete work, translated in full from the original Russian, 217 pages, typeset to be read rather than reprinted.

First faithful English translation from the original Russian edition.

Translated by Valery Manokhin, PhD. Northern Star Academic Press.

The full series: russianmathbooks.com

Sources: MacTutor History of Mathematics (biography of Nikolai Fuss); the Euler Archive’s translation of Fuss’s 1783 Eulogy of Euler by John S. D. Glaus; Wikipedia on Nicolas Fuss and Paul Heinrich Fuss; the 1820, 1823 and later Russian catalogue records of the three parts.

Imagine a mathematics lesson in which the teacher asks you to guess the answer.

You choose eight. It is wrong.

You choose seven. That is wrong too.

But instead of asking you to try again, the teacher tells you that you now have enough information to calculate the exact answer.

You do not need a third guess. You need to understand the first two mistakes.

This is the idea behind double false position, a method that allowed people to solve linear problems using ordinary arithmetic rather than the symbolic equations we would usually write today. It belonged to a substantial mathematical tradition, not a collection of party tricks.

And its disappearance from arithmetic teaching was noticed at the time.

On 6 September 1877, a contributor signing as “T. H.” published a discussion of double position in the New England Journal of Education. The writer described the method as:

“so unwisely discarded from modern arithmetics”

Here, “arithmetics” meant arithmetic textbooks. Almost a century and a half ago, someone was already arguing that this particular act of educational modernisation had thrown away something worth keeping.

After looking closely at the method, I think there is a good case for listening.

The Russian page that caught my attention

I recently came across a striking example in a Russian account of old arithmetic manuscripts.

The passage calls the procedure the “false rule” and explains how the calculator records the errors produced by assumed answers. The name sounds suspicious until you understand it: the rule is not false. The starting assumptions are.

The problem is simple enough to state without algebra:

Find a number which, when multiplied by fourteen and divided by four and two-thirds, gives eighteen.

The first trial number is 8.

Following the instructions produces 24, which is 6 too much.

The second trial number is 7.

This produces 21, which is 3 too much.

The manuscript’s calculation arranges the trial numbers and their errors around a crossed-line diagram. Each trial number is multiplied by the error belonging to the other trial:

7×6=42,8×3=24.

Because both results are too large, the calculator subtracts the cross-products, then divides by the difference between the errors:

Answer=42−246−3=183=6.

The two wrong answers have produced the right one.

Checking is straightforward: six multiplied by fourteen is eighty-four, and eighty-four divided by four and two-thirds is eighteen.

There is a lovely additional detail in the Russian passage. A cross-like symbol marks an excess, while a symbol resembling our modern division sign marks a deficit. In this notation, the division-shaped symbol does not instruct the reader to divide. It means “too little.”

The diagram records not merely that an answer is wrong, but how it is wrong.

That distinction is the whole method.

A mistake can contain a measurement

At first glance, the cross-multiplication looks like an arbitrary recipe.

Why should multiplying the wrong numbers by the wrong errors produce anything useful?

Look again at the two trials.

Reducing the input from eight to seven reduces the output from twenty-four to twenty-one.

A decrease of one in the input causes a decrease of three in the output.

At seven, the output is still three too large. Therefore, reducing the input by one more unit removes the remaining excess.

The answer must be six.

The crossed calculation is a compact way of carrying out that reasoning.

In modern mathematical language, the two trials reveal the slope: how rapidly the result changes when the input changes. The error tells us how far the output must move; the slope tells us how much to change the input.

For this example:

Correction=remaining output erroroutput change per unit of input=33=1.

So we move from seven to six.

Notice something else: the two guesses do not have to surround the answer. Seven and eight are both too large. That does not matter, because the relationship is a straight line and we can extend it beyond the two points.

This is not blind trial and error. Blind trial and error would keep guessing until something worked.

Double false position uses the trials to discover a relationship, then uses that relationship to calculate the answer. Historical practitioners could interpolate between two trials or extrapolate beyond them. (ResearchGate)

Why bother, when algebra makes this so easy?

There is an obvious objection.

The Russian problem, written in modern notation, is simply

14x423=18.

Since four and two-thirds is fourteen-thirds, the left-hand side reduces to 3x. The answer is immediately x=6.

Why make two guesses and perform cross-multiplications when one line of algebra will do?

For this particular problem, there is no efficiency advantage. Even a proportional correction from the first trial would be enough: eight produces twenty-four, so the number producing eighteen is 8×18/24=6.

But double false position is not limited to direct proportion. It also works when the calculation includes a fixed amount added or subtracted.

Consider a modern illustration.

Suppose a printer charges a fixed setup fee plus a constant price per copy. You have two quotations: twenty copies cost £90, while fifty copies cost £180.

How many copies can you order for exactly £150?

The twenty-copy quotation is £60 below your target. The fifty-copy quotation is £30 above it.

The two errors are now on opposite sides, so the corresponding calculation adds rather than subtracts:

Number of copies=20×30+50×6030+60=360090=40.

You have found the answer without first calculating either the setup fee or the price per copy.

The intuitive explanation is just as clear. An additional thirty copies cost an additional £90, so each extra copy costs £3. Moving from the £90 quotation to the £150 target requires another £60, or twenty more copies.

Starting with twenty copies, you arrive at forty.

The fixed setup fee prevents you from simply scaling the original quotation. But it disappears when you compare the two quotations.

The difference between two trials reveals information that either trial alone does not provide.

That is a mathematical idea worth understanding, even after you have mastered algebra.

This was an international mathematical tradition

The Russian example was not an isolated invention.

Double-false-position techniques appear in ancient Chinese mathematics and medieval Arabic mathematical writing. In the Arabic tradition, the method was known as hisāb al-khaṭāʾayn, referring to reckoning with two errors. Fibonacci discussed it in Liber Abaci, drawing on Arabic sources. The earlier relationships between the Chinese, Indian and Arabic traditions are more complicated than a simple story of one culture handing the method to another.

A particularly tangible example survives in the manuscript of Michael of Rhodes, a fifteenth-century mariner whose mathematics is documented by the Michael of Rhodes Project.

Michael solves a problem about the value of a cargo of pepper using several methods, including double false position. His manuscript uses a large X-shaped arrangement to organise the trial values, the discrepancies and the cross-multiplications.

That is recognisably the same kind of calculating aid as the crossed diagram in the Russian passage.

The notation varies. The language varies. The structure remains familiar: assume a value, calculate the consequence, record the discrepancy, and combine it with another trial.

These examples also correct a tempting misconception. False position was not merely something people used before algebra existed. Michael knew algebra and used both approaches. The methods could coexist as alternative tools.

Knowing one way to solve a problem did not make every other way worthless.

What happened in the nineteenth century?

The historical claim needs to be stated carefully.

There was no single worldwide date on which schools stopped teaching false position. The evidence does not support a neat announcement that, sometime in the nineteenth century, everyone abandoned it.

What we can document is that its removal from arithmetic textbooks was already being criticised in 1877. That contemporary complaint is more useful than a sweeping claim about every country’s curriculum.

The attraction of replacing it with algebra is easy to understand.

A problem that might previously require a named rule and an arrangement of numerical trials can be expressed as

ax+b=T,

then solved generally:

x=T−ba.

That is a powerful improvement. It expresses the structure directly, works with arbitrary coefficients, and fits into a much larger mathematical language.

On the narrow question of how quickly to solve a straightforward linear equation, algebra usually wins.

But there are two different educational questions here:

What is the shortest method for obtaining the answer?

And what does each method help the student understand?

They do not always have the same answer.

The crossed-line recipe, taught mechanically, could certainly become just another rule to memorise. There is nothing inherently enlightening about multiplying diagonally because a textbook says so.

But explained properly, the method provides a concrete introduction to rate of change, linearity, interpolation and the use of error as information.

By 1967, The Mathematics Teacher was presenting double false position in a column titled “Historically Speaking.” The technique was still worth discussing, but now explicitly as a historical subject.

I would argue that it deserves a little more than historical curiosity.

The idea did not disappear from mathematics

There is an important twist: the old arithmetic method is closely related to something still taught in numerical mathematics.

Suppose we define the error in our Russian example as

e(x)=3x−18.

The two trials give us two points on the error graph: (8,6) and (7,3).

Draw the straight line through those points and find where it crosses zero. That crossing occurs at x=6.

This is the same straight-line zero-finding calculation used in a secant step. The secant method uses two function evaluations to estimate a slope, then uses that slope to calculate a new estimate of a solution.

In the historical linear problem, the line through the two trials is not an approximation to the relationship. It is the relationship. That is why the answer is exact.

For a nonlinear problem, the situation changes. Two points define a line, but the actual relationship may curve away from it. The calculated crossing is then generally only an approximation, and further iterations may be required. The modern regula falsi algorithm uses the same interpolation idea while preserving two endpoints that bracket a root.

So the method was not rendered mathematically meaningless when symbolic algebra became the preferred way of solving elementary equations.

Its underlying calculation also belongs to numerical methods for problems where a convenient symbolic solution may not be available.

And what about backpropagation?

Working in machine learning, it is natural to recognise a resemblance: make a trial, calculate an error, use information about that error to guide an adjustment.

But it would be misleading to call the old crossed diagram an early form of backpropagation.

Backpropagation computes derivatives by applying the chain rule backward through a computation. An optimiser can then use those derivatives to update parameters. Double false position instead infers a slope from two complete trial calculations; it does not propagate derivatives backward through the individual operations.

The conceptual resemblance is real, but the mechanisms are different.

The stronger mathematical connection is to the secant method, not to neural-network training.

The historical method does not need an exaggerated claim about artificial intelligence to be interesting. It already demonstrates something fundamental: a carefully measured error can tell you how to move towards a solution.

What I would bring back to the classroom

I would not replace algebra with false position.

Nor would I require students to memorise an antique cross-multiplication procedure merely because it is old.

I would put the Russian example in front of them before giving them the formula.

Eight produces twenty-four. Seven produces twenty-one. We need eighteen.

What do those two trials tell us?

Then I would ask why the answer can lie outside the two guesses. What changes when one trial is too high and the other too low? Why does adding a fixed charge not break the method? What happens when the calculation involves squaring the input?

Those questions turn a historical procedure into an investigation.

They also reveal its limits. The exact rule depends on a nonconstant straight-line relationship and exact arithmetic. Two trials are not enough to identify an arbitrary curve, and real measurements may contain noise.

Understanding why a method works includes understanding when it does not.

For me, that is the case for recovering this piece of arithmetic: not nostalgia, and not the belief that older textbooks were automatically better, but the opportunity to approach familiar mathematics from an unfamiliar direction.

Algebra lets us manipulate a relationship symbolically. False position lets us interrogate it numerically.

Both perspectives are valuable.

The nineteenth-century writer who regretted the method’s disappearance had a point. We gained a more efficient symbolic language. That does not mean there was nothing left to learn from the method it displaced.

Sometimes a wrong answer is not something to erase. Once you know precisely how wrong it is, it becomes part of the solution.

I thought I knew where the modern Russian mathematics textbook began.

The obvious answer was Leonty Magnitsky’s Arithmetic, published in Moscow in 1703. It was the monumental mathematical course that became one of the foundational books of Russian education.

Then I discovered a much smaller book that changes the opening chapter of that history.

On 15 April 1699 — four years before Magnitsky — Ilya Kopievsky published Краткое и полезное руковедение во аритметику: A Brief and Useful Guide to Arithmetic.

It contained only 48 pages. The surviving copy held by the Russian State Library measures approximately 15 by 9.5 centimetres. And it was printed not in Moscow, but in Amsterdam, at the press of the Dutch merchant Jan Thessing.

Its reported print run was 3,350 copies — remarkably, more than the 2,400 copies later printed of Magnitsky’s vastly more substantial Arithmetic. As of 2022, researchers had identified only seven surviving copies of Kopievsky’s little book.

This does not dethrone Magnitsky.

It changes the prologue.

Which book was really first?

The question depends on what we mean by “first.”

In 1682, the Moscow Print Yard had already published Считание удобное, or Convenient Reckoning. But that work was essentially a printed multiplication table, not a systematic arithmetic course.

Kopievsky’s 1699 Guide was the first recognisable printed arithmetic teaching manual intended for Russian readers.

Magnitsky’s 1703 Arithmetic was something on an entirely different scale: a broad, ambitious and comprehensive mathematics curriculum. It was the book that successfully established the model and entered the national canon.

The accurate story, therefore, is not that Magnitsky falsely received credit for somebody else’s achievement.

The more interesting story is that Magnitsky’s famous textbook had a nearly forgotten precursor — and that this precursor had to be produced outside Russia.

Peter’s information-infrastructure problem

Peter the Great’s reforms are normally told through physical things: ships, cannon, shipyards, factories, canals and cities.

But modernisation also required information.

A state could order the construction of a navy, but it could not train navigators without books.

It could establish technical schools, but those schools required textbooks.

It could invite foreign engineers, but Russian students still needed access to mathematics, astronomy, navigation, architecture, military science and foreign languages.

Peter’s reforms therefore faced what we would now call an information-infrastructure or knowledge-supply problem.

During the Great Embassy of 1697–1698, the Dutch Republic offered an obvious solution. Amsterdam was one of Europe’s great centres of publishing, with capital, skilled printers, international trade connections and books appearing in dozens of languages. Peter and his circle already knew Amsterdam publishing, and the tsar’s own library contained numerous Dutch editions.

Moscow possessed considerable printing capacity. The obstacle was not simply the physical absence of printing presses.

The Moscow Print Yard produced books almost exclusively on religious subjects, normally with the patriarch’s blessing. Producing an entire series of secular textbooks based on the works of non-Orthodox European scholars would have been institutionally awkward.

There was an even more fundamental problem: late-seventeenth-century Moscow did not possess the necessary library of foreign scientific literature from which such textbooks could be assembled.

Amsterdam had both the books and the production system.

So Peter effectively outsourced the first stage of his secular educational programme.

A seventeenth-century publishing start-up

The venture began through Jan Thessing, an Amsterdam merchant whose family had long traded with Russia.

In May 1698, while Peter was travelling from Amsterdam towards Vienna, he approved Thessing’s proposal to manufacture and export Russian-language printed material. The formal charter followed in February 1700, but publication had already begun.

Thessing received a 15-year privilege to print and sell maps, diagrams, portraits, mathematical and architectural works, and books concerning military subjects. The charter said that the publications should serve the public benefit and the teaching of arts and knowledge.

One restriction is particularly revealing: religious books were excluded. Those remained the domain of the Moscow Print Yard.

Rather than immediately transform the old ecclesiastical publishing machinery, Peter created a parallel channel for secular knowledge.

In modern terms, the arrangement resembled an international publishing start-up.

Peter supplied political backing, demand and a privileged market.

Thessing supplied capital, commercial connections, equipment and distribution.

But they still needed somebody capable of converting European knowledge into books that Russian students could read.

That person was Ilya Kopievsky.

The man between worlds

Calling Kopievsky a “Russified Pole” is tempting, but far too simple.

He appears to have been born around 1651 into a minor Calvinist noble family in the Grand Duchy of Lithuania, probably somewhere in the territory of present-day Belarus. His exact birthplace remains uncertain.

When he was eight or nine years old, during the Russo-Polish War, he was captured by forces associated with Prince Ivan Khovansky and taken into Muscovy. He remained there for approximately six years before returning home.

His troubles did not end there. His family estate was confiscated, leaving him without the position or income he might otherwise have inherited.

Kopievsky studied and later taught at the Protestant school in Slutsk. Eventually, he moved to Amsterdam and became connected with the Dutch Reformed Church. In his writings he sometimes described himself as Polonus — a Pole — but his linguistic, political and cultural identity crossed several modern national categories.

That complicated background made him almost uniquely suited to Peter’s project.

He knew the Slavic literary traditions used by Russian readers.

He spoke several European languages.

He had experience as a teacher.

He understood Protestant educational practice.

And he was living in the greatest multilingual publishing centre in Europe just as Peter’s representatives arrived looking for practical knowledge.

Kopievsky began teaching Russian students foreign languages, navigation-related subjects and other “arts and sciences.” Crucially, he was teaching them from instructional materials that he was simultaneously writing or compiling himself.

Thessing may have owned the enterprise, but Kopievsky became its intellectual engine.

He was simultaneously an author, translator, editor, curriculum designer, terminology builder, typographic organiser and teacher. Recent scholarship suggests that he determined almost the entire publishing programme himself.

Between March 1699 and the middle of 1700, he prepared a remarkable series of textbooks dealing with world history, arithmetic, astronomy, military science, Latin and other foreign languages, as well as Aesop’s fables.

The title pages invoked Peter’s authority, but Kopievsky was not merely a clerk mechanically executing a list supplied by the tsar. He was an intellectual entrepreneur constructing the curriculum largely in real time.

The strangest 48-page arithmetic textbook

Kopievsky’s Brief and Useful Guide to Arithmetic appeared on 15 April 1699.

Its full title promised instruction in “every kind of reckoning in the counting of all things.”

But the contents are not what a modern reader would expect.

Only 16 of its 48 pages are actually devoted to arithmetic.

The mathematical section covers five principal subjects:

Numbers.

Addition, called придача.

Subtraction, called убавка.

Multiplication.

Division.

Kopievsky introduces Arabic numerals, the older Cyrillic system of representing numbers and Roman numerals. Zero is called ничево — an old spelling of “nothing.”

The mathematical terminology is particularly revealing. Imported European words appear beside Slavic alternatives:

Аддиция alongside придача.

Мультипликация alongside умножение.

Дивизия alongside разделение.

The page looks like a competition between possible mathematical languages, with Kopievsky testing imported and native terms side by side.

As a mathematics textbook, it was extremely rough.

There are very few worked examples.

There are no exercises for the reader.

Multiplication and division are explained inadequately.

At least one of the handful of printed sums is simply wrong: the book gives 5,324 + 3,063 as 8,386 rather than 8,387.

Other apparent typographical errors occur in the examples.

This was not a polished masterpiece.

It was a prototype produced under extraordinary pressure.

Arithmetic, morality and Aesop’s fables

The strangest feature of the book is what fills the remaining pages.

Kopievsky included moral sayings, rhymed instructions and no fewer than 17 fables.

Why place Aesop inside an arithmetic manual?

Part of the answer appears to lie in Dutch educational practice, where fables and moral instruction commonly accompanied elementary teaching. Kopievsky was not merely trying to communicate methods of calculation. He was adapting a broader European model of schooling for a new audience.

The first fable is particularly appropriate.

A hen, scratching through rubbish, discovers a diamond.

A merchant or jeweller would have understood its enormous value. The hen does not. It would rather find a grain of wheat or barley.

Kopievsky explains that the diamond represents wisdom and education. Those who do not understand the liberal arts resemble the hen: they encounter something precious but cannot recognise its value.

The fable reads almost like a manifesto for Peter’s educational programme.

Russia did not merely require new books.

It required readers capable of recognising why the knowledge inside them mattered.

And three centuries later, the fable has acquired an unintended second meaning. Kopievsky’s own little book became something like the discarded diamond in his story — overshadowed by a greater textbook and almost forgotten.

Inventing scientific Russian in real time

It would be anachronistic to describe Kopievsky’s language as modern Russian.

The National Electronic Library catalogues the book as Church Slavonic. Linguistic research describes Kopievsky’s writing more precisely as a mixture of Russian and Church Slavonic, containing Ukrainian, Polish and other European influences.

The instability of the language is not a defect incidental to the story.

It is one of the most historically important things about the book.

Kopievsky had few established models for expressing European scientific and educational terminology in a language suitable for Russian readers. He therefore had to locate, adapt, combine and sometimes invent equivalents.

A recent biographical study argues that he was effectively helping to invent a Russian language of science.

This explains why imported and native terms coexist so visibly.

There was no settled vocabulary to consult.

There was no comprehensive editorial style guide.

There was no established secular textbook industry.

There was one overworked multilingual teacher in Amsterdam attempting to construct all three at once.

According to Kopievsky’s own account, the work became exhausting. He was writing books, organising the printing operation and teaching inadequately trained workers how to produce Cyrillic editions.

His complaints may have contained some self-dramatisation, but the physical evidence supports the basic picture: these books were produced rapidly, experimentally and under difficult conditions.

Why Magnitsky won

Four years later, Magnitsky’s Arithmetic appeared in Moscow.

It was larger, deeper, more coherent and far better produced. It offered not merely elementary reckoning but a broad mathematical curriculum.

Contemporary and modern assessments agree that it greatly surpassed Kopievsky’s booklet in volume, content and printing quality. Although Kopievsky’s reported print run was larger, print run alone is not a measure of influence.

Thin schoolbooks are heavily used and easily destroyed. The Amsterdam volumes were not particularly well printed, and their transportation and distribution inside Russia remain incompletely documented. Magnitsky’s substantial volume was more valuable to owners, more likely to be bound and more likely to survive.

Magnitsky therefore deserves his canonical position.

But he did not appear in a vacuum.

Kopievsky’s book demonstrates that an effort to create a new kind of mathematical education had already begun. The first experiment was smaller, weaker and produced abroad — but it established a precedent.

The proper conclusion is not that Magnitsky was wrongly celebrated.

It is that textbook history tends to remember successful monuments and forget the prototypes that made them imaginable.

Magnitsky built the monument.

Kopievsky built the prototype.

The man who could not afford his own burial

The Amsterdam enterprise did not last.

Kopievsky believed that Thessing underpaid him and failed to recognise the scale of his contribution. Their partnership collapsed. A subsequent venture with another Dutch merchant also ended in conflict and litigation.

Kopievsky carried his publishing ambitions, manuscripts and typographic equipment through different parts of Europe, searching for patrons and commercial partners.

Eventually, he entered Russian service as a translator and moved to Moscow.

There, far from becoming a celebrated founder of Russian secular publishing, he disappeared into relative obscurity. He died in September 1714 in poverty.

His widow petitioned the authorities for assistance. She reported that he had left numerous debts and that the household did not possess enough money even to bury him.

It is a brutal conclusion to the life of a man who had helped manufacture some of the earliest secular textbooks for Peter’s Russia.

And it brings us back to the hen and the diamond.

Kopievsky’s arithmetic was not a masterpiece. Its mathematical content was thin. Its presentation was hurried. Its examples contained mistakes.

But it captured something more important than perfection.

It captured the moment when a new educational culture was being assembled: a new publishing system, a new secular curriculum and even a new scientific vocabulary.

That culture was not created entirely inside Russia.

It emerged through an international network: a Russian tsar, a Dutch merchant, Amsterdam printers, European scientific books and a Calvinist intellectual from the borderlands of the Grand Duchy of Lithuania.

The modern Russian mathematics textbook began not as an exclusively national creation, but as a transnational publishing experiment.

Before Peter’s Russia could print its new future at home, it printed a prototype of that future in Amsterdam.

The Polytechnic Museum in Moscow holds object number 12402.

It is a metal ovoid, 410 millimetres long, hand-beaten from aluminium sheet. You can see the weld seams. The underside is painted in the grey hammertone crackle enamel that every Soviet instrument workshop used, and several of the screw heads are sealed with red lacquer — the tamper-evident paint they put on things that were not meant to be opened.

The catalogue calls it Кибернетическая модель «Черепаха». The cybernetic model “Turtle”.

Creation date: 1956–1957. Creation place: Институт автоматики и телемеханики АН СССР — the Institute of Automation and Remote Control of the USSR Academy of Sciences.

The museum says it is the only surviving specimen.

What it does

There is a photocell behind a chrome bezel in the nose. That is the eye.

With nothing in front of it, the turtle drives toward a light source. When a shadow falls across the photocell, an obstacle routine fires — it reverses, turns aside, and carries on. There is a microphone under a perforated grille on its back. Sound a tone and it stops.

So far this is a reflex machine. Percept in, action out. Nothing more.

Then the museum’s own description says this:

Если включать сигнал каждый раз, когда «черепаха» наезжает на препятствие, срабатывает её способность обучаться — тогда, «услышав» сигнал, она не просто тормозит, а сразу начинает обходить преграду, ещё не подъехав к препятствию вплотную.

Sound the tone every time it hits something. After enough repetitions its capacity to learn kicks in — and now, on hearing the tone, it does not merely brake. It begins the avoidance manoeuvre immediately, before it has reached the obstacle at all.

That is Pavlovian conditioning. In relay-contact logic. In 1956.

And it is worth being precise about what changed, because it is the whole of the thing. Before conditioning, the machine acts on what it senses. After conditioning, it acts on what it expects. The tone carries no information about obstacles — it acquires that information by co-occurrence, and the machine begins behaving on the strength of an association it built itself.

Percept to prediction. That is what a learned model is. Everything since has been a question of how you represent the association and how you fit it.

Now the parts that stop this being a good story

Two things, and I would rather say them myself than have them said to me.

First: Grey Walter got there first. Elmer and Elsie were built in Bristol in 1948–49 and publicised by 1951, and Walter’s Machina docilis did conditioning with his CORA circuit. The Moscow turtle is seven to eight years later. This is not a priority claim and I am not making one.

Second: it has a mains plug.

Look at the photograph. That is a cloth-braided flex with a red tracer, ending in a two-pin Soviet plug. There is no battery compartment. There is no charging arrangement.

Walter’s tortoises were battery-powered, and their most celebrated behaviour was finding their way back to a hutch to recharge when they ran low. The Moscow machine is tethered. It works within the radius of its cable. On autonomy — the axis that made Walter famous — it is the lesser machine, and the photograph proves it.

There is one genuine engineering difference in the other direction. Walter built with thermionic valves. This is built with relay contacts, which was the Soviet strength — Tsypkin and Gavrilov territory, an entire school of switching theory. And the face carries two fixed apertures where Walter used a single rotating photocell. Two fixed sensors suggests differential sensing: compare left and right, steer toward the brighter. Simpler, no moving scanner. The photographs cannot settle it. The museum’s accession file might.

Why 1956 is the interesting number

Not because it was first. Because of what it was late to.

Cybernetics was a banned pseudoscience in the Soviet Union until the mid–1950s. The turning point is usually dated to 1955, when Sobolev, Kitov and Lyapunov published «Основные черты кибернетики» in Вопросы философии. Before that, the word itself was an accusation.

The turtle was built in the eighteen months after that. Somebody at the Institute of Automation and Remote Control read the paper that said this was allowed now, went into the workshop, and built a machine that learns.

That is the story. Not who was first. What it looks like when a field is switched back on.

Who built it

The museum’s author field is empty. No maker’s plate appears in any of the three published photographs.

But the Institute of Control Sciences — the same institute, renamed in 1969 — says on its own memorial page for Alexander Mikhailovich Petrovsky (1925–1993) that he built a «кибернетическая черепаха», one that «передвигалась, обходила препятствия, имитировала некоторые условные рефлексы и обучалась несложным командам». Moved, avoided obstacles, imitated conditioned reflexes, learned simple commands.

Four for four, at the same institute, in the same period.

The identification is not documented. It is very likely, and I will say so in exactly those words until someone opens the accession file.

Petrovsky is worth knowing about regardless. His first published paper, three years before the turtle, describes a white-noise generator built from a rotating drum full of steel ball bearings, with a voltage across the axis and the drum wall. The balls tumble, the contact resistance jitters, the current fluctuates. He models the ball falls as a Poisson process and derives a Lorentzian spectrum with a corner frequency of NF — number of balls times rotation rate.

You tune the bandwidth of the noise source by counting ball bearings.

His grandfather was chief sanitary doctor of Moscow. His father was a Formalist literary theorist who ran the theoretical poetics section at GAKhN and whose translation of Manon Lescaut is still considered unsurpassed. His mother steered him into engineering deliberately, on the reasoning that the country needed technical people.

He was thirty-one when the turtle was built.

The part that makes this a machine learning story

Petrovsky founded Laboratory №38 at that institute in 1968.

In 1971 the laboratory of A. Ya. Lerner was struck from the institute’s structure. His people were about to be scattered.

Petrovsky’s laboratory took them in. Among them: Vladimir Vapnik and Alexey Chervonenkis.

The institute’s own history is blunt about how unusual this was. Moving between laboratories was categorically discouraged, and the directorate’s preferred solution to a conflict was to be rid of everyone involved. Lab 38 was the standing exception — it absorbed Feldbaum’s orphaned group in 1968, Vapnik and Chervonenkis in 1971, and Lyubinsky’s neuron-modelling people in 1977 when their laboratory head died and their lab was struck out too.

Statistical learning theory has a continuous institutional home in Moscow from 1971 onward, and the reason is a man who built a turtle out of aluminium sheet and a noise generator out of ball bearings.

Laboratory 38 still exists. It is now «Управления по неполным данным» — control under incomplete data. One of its four stated research directions is machine learning and pattern recognition, described as building on Vapnik–Chervonenkis foundations.

Sixty-nine years after the turtle. Same laboratory. Same building.

The turtle is object 12402 at the Polytechnic Museum, Moscow. Petrovsky’s 1953 paper is «Генератор флюктуационных шумов для инфразвуковых частот», Автоматика и телемеханика XIV:4, 440–444 — free full text at Math-Net.Ru, paper id at13672. It has been downloaded 129 times.

I am working through the Soviet machine learning literature paper by paper — what was done, what was rediscovered later in the West, and what is still sitting there unread. If that is your kind of thing, follow along.

In today’s AI-driven world, we casually speak of generalization, overfitting, and VC dimensions — as if these ideas were always part of the machine learning landscape. But their origins are remarkably precise, and their creators remarkably overlooked.

Meet Vladimir Vapnik and Alexey Chervonenkis, two Soviet mathematicians who built the mathematical foundation of machine learning — at a time when “AI” was mostly philosophical and “big data” didn’t exist.

🎓 Who Were They?

Vladimir Vapnik

• Born: 1936, Soviet Union
• Education:

Undergraduate: Mathematics and Physics, Uzbek State University (Tashkent)

PhD: Institute of Control Sciences, USSR Academy of Sciences

Discipline: Mathematical statistics, information theory, pattern recognition

Profession:

• Worked at the Institute of Control Sciences in Moscow (a major Soviet research hub in applied math and engineering)
• Later joined AT&T Bell Labs in the U.S. (1990s), where he further developed Support Vector Machines

Despite his later association with statistical theory, Vapnik was never trained or employed as a statistician. His work was grounded in mathematics, physics, and control theory — disciplines concerned with information, systems, and optimization.

Alexey Chervonenkis

• Born: 1938, Soviet Union
• Education:

Undergraduate and graduate degrees in mathematics, Moscow Institute of Physics and Technology (department of Radiotechnics and Cybernetics)

Discipline: Probability theory, mathematical modeling, pattern recognition

Profession:Researcher at the Institute of Control Sciences, where he began his long collaboration with Vapnik. Later taught at Moscow Institute of Physics and Technology (MIPT)

Chervonenkis was a quiet but profound thinker. He was never part of the Soviet statistics community; his orientation was always in theoretical computer science and applied mathematics.

🧠 What They Did — and Why It Matters

In 1964, the two researchers published their breakthrough paper introducing VC theory (Vapnik–Chervonenkis theory). This work asked — and answered — one of the most fundamental questions in learning:

How can we know if a machine will perform well not just on the training data, but on unseen examples?

They formalized:

• Generalization bounds
• Capacity control (via VC dimension)
• Uniform convergence
• Empirical risk minimization (ERM)

This wasn’t just a mathematical curiosity. These tools became the core principles behind supervised learning — they explained how and why learning from data could work at all.

In essence, Vapnik and Chervonenkis gave machine learning its first rigorous statistical soul — without coming from statistics themselves.

❌ Not Statisticians — And That’s Important

It’s a historical misunderstanding to attribute the birth of machine learning to traditional statisticians.

• Vapnik and Chervonenkis did not publish in mainstream statistical journals.
• They weren’t interested in estimation or hypothesis testing in the classical sense.
• They worked on learning theory: a new field, with its own rules, concerns, and mathematics.

In fact, many in the Western statistical community ignored or misunderstood their work for decades. It was only when Vapnik moved to the U.S. and helped develop non linear version of Support Vector Machines in the 1990s that his contributions began to be widely recognized. The original liner version of Support Vector Machines we already developed by Vapnik long before that in the early 1960s in the USSR, it was known under the name of ‘The Methods of Generalised Portraits.’ Thirty years later, whilst working in Bell Labs in the USA Vapnik and Cortes combined his linear Support Vector Machine method with kernel methods (invented by Aizerman and his colleagues like Rozonoer from the Aizerman’s Lab in late 1950s-early 1960 in the same institute that Vapnik has worked in the USSR)

🇺🇸🇷🇺 Two Nations, One Legacy

Machine learning, as a field, has only two true birthplaces: the United States and the Soviet Union.

• The U.S. gave us early systems like Arthur Samuel’s checkers-playing program (1959) and Frank Rosenblatt’sPerceptron (1958).
• The USSR gave us the mathematical foundations through Vapnik and Chervonenkis, beginning in the early 1960s.

Every country since has contributed to scaling, refining, and applying machine learning — but only the USA and USSR can claim to have founded it.

📘 A Legacy That Shaped the Future

When Vapnik published The Nature of Statistical Learning Theory in 1995, it was more than a summary of his research. It was a global call to see learning as a mathematical discipline, not just a computational trick. The book introduced generations of researchers to the principles behind SVMs, generalization, and capacity — ideas that now live inside nearly every ML algorithm.

Even today, the explosion of large language models, deep networks, and reinforcement learning is still shaped by the core insights Vapnik and Chervonenkis provided:

You must control complexity to achieve generalization.

🏛️ The Founding Fathers of Machine Learning

To call them “influential” is an understatement.

Vladimir Vapnik and Alexey Chervonenkis founded the theory that makes machine learning not just possible, but principled.
They weren’t statisticians. They weren’t riding a trend.
They invented the discipline before most people even knew it existed.

If machine learning is a cathedral, then Vapnik and Chervonenkis laid its mathematical cornerstone — quietly, brilliantly, and decades before the rest of the world showed up to build on it.

Why Kiselev?
Why Rachinsky?
Why Malinin and Burenin?
Why old arithmetic, algebra, geometry, and problem books from the Russian Empire and early Soviet period?

The answer is simple: because many of these books were excellent.

They were clear.
They were rigorous.
They respected the child’s mind.
They did not replace mathematics with entertainment.
They did not confuse pedagogy with fashion.
They did not treat children as incapable of serious thought.

And there is a sad historical story behind this.

The USSR inherited one of the strongest mathematical textbook traditions in the world. Then, under Khrushchev and later Soviet reformers, it began to damage that inheritance.

This was not progress.

It was demolition dressed up as modernisation.

The level before the demolition

This is the part people often miss.

Before Soviet reforms damaged the old textbook tradition, Russian school education was already operating at a level that would be almost unimaginable today.

In the classical gymnasiums, Russian schoolchildren could encounter calculus around the age of 14. They were not merely doing arithmetic forever. They were being prepared for serious algebra, geometry, analysis, and scientific thinking.

At roughly the same historical period, many American schoolchildren were still barely finishing arithmetic.

That contrast matters.

Russia did not suddenly become strong in mathematics because of some abstract Soviet miracle. The foundations were older. They came from the imperial Russian gymnasium tradition, serious textbooks, demanding teachers, and a culture that treated mathematics as intellectual formation rather than classroom decoration.

This is why the destruction of the old textbook tradition was so damaging.

It was not just a change of books.

It was the dismantling of an educational ladder that had already taken children from arithmetic into serious mathematics at an age when other systems were still teaching the basics.

The old Russian textbook tradition

Before the reforms, Russia had a remarkable school textbook culture.

Kiselev is the obvious example.

His books in arithmetic, algebra, and geometry became legendary because they did exactly what school mathematics books are supposed to do: build the mind step by step.

They were not flashy.
They were not full of bureaucratic “activities.”
They were not written to satisfy committees.
They were written to teach.

A good mathematics textbook is not a pile of colourful pages. It is not a psychological experiment conducted on children. It is not a collection of disconnected tricks.

A good mathematics textbook is a ladder.

First the child learns arithmetic properly.
Then algebra begins to make sense.
Then geometry teaches proof.
Then higher mathematics is no longer an alien planet.

The old Russian books understood this.

They trained calculation, attention, memory, abstraction, proof, and reasoning. They assumed that children could think. They did not apologise for difficulty. They did not water mathematics down into slogans.

This is why these books survived for generations.

Kiselev’s books became part of the backbone of Russian mathematical schooling because they worked.

Then came Khrushchev

The tragic irony is that the Soviet Union inherited this great Russian educational tradition and then began damaging it from within.

Under Khrushchev, the Soviet system entered a period of aggressive educational reform.

Khrushchev is often described through his Ukrainian political biography: he rose through the Ukrainian party apparatus, became head of the Ukrainian party in 1938, and was deeply connected with Ukraine throughout his career.

The safe point is this: Khrushchev was not the guardian of the older Russian gymnasium tradition; he was the Soviet political boss under whom a new reform climate began.

That distinction matters.

The Soviet Union was not identical with the older Russian tradition. In some areas it inherited, expanded, and exploited that tradition. In others it damaged it.

School mathematics is one of those tragic cases.

The USSR had received a powerful inheritance: imperial Russian gymnasium culture, serious textbooks, demanding teacher training, and a clear ladder from arithmetic to algebra to geometry and beyond.

Then the reformers arrived.

Markushevich before Kolmogorov

This is where the story is often told incorrectly.

People jump straight to Kolmogorov.

Yes, Kolmogorov later became associated with the Soviet school mathematics reform. Yes, his authority mattered. Yes, his name became attached to the later stage of the reform.

But he was not where the demolition began.

The earlier reform current was already moving against the old textbook tradition. In the Russian source I am drawing on, the central early figure is Alexey Ivanovich Markushevich.

The source is extremely harsh about him:

“The leader of the reformers of school mathematical education, Alexey Ivanovich Markushevich, did not distinguish himself by special achievements in scientific activity, but shone in the para-scientific field: he abolished Kiselev’s brilliant method…”

That is the key point.

The demolition of Kiselev should not be lazily framed as “Kolmogorov did it.”

Kolmogorov came later.

Markushevich was already a major educational bureaucrat and mathematics educator before the later Kolmogorov-branded reform. Standard biographies place him at the centre of Soviet/Russian educational institutions: he was vice-president of the Academy of Pedagogical Sciences of the RSFSR, deputy minister of education of the RSFSR from 1958 to 1964, and later vice-president of the USSR Academy of Pedagogical Sciences.

What kind of people wrote the new textbooks?

The Russian source is bitter for a reason.

It does not present the replacement of Kiselev as some innocent pedagogical update. It presents it as an act carried out by people of a very different calibre from the old school textbook builders.

The same passage continues with an even more poisonous accusation against Markushevich:

“…and he turned out to be the main buyer of medieval European manuscripts stolen from the Central State Archive of Ancient Acts. This is the calibre of people who have been writing textbooks for our children since the seventies…”

This is a savage line, but the manuscript scandal is not merely invented internet abuse. Standard biographical summaries also mention that Markushevich’s final years were overshadowed by a scandal involving the theft of medieval European manuscripts from RGADA, the Russian State Archive of Ancient Documents, and that investigations identified him as the main buyer of the stolen documents.

That does not by itself prove that every textbook he touched was bad.

But it does show why the Russian polemical account is so angry.

The point is not simply “old good, new bad.”

The point is that the old Russian textbook tradition was built by teachers and textbook writers who understood the classroom. Then a new class of reformers, administrators, and famous names began replacing a proven method with abstract schemes and institutional authority.

The result was predictable.

The attack on Kiselev

Kiselev is the symbol of this loss.

His books had educated enormous numbers of students. They were part of the backbone of Russian mathematical schooling. They had clarity, order, and intellectual seriousness.

But instead of being treated as a national educational treasure, they were pushed aside.

The older textbook tradition was attacked precisely because it was old. It belonged to a pre-reform world. It carried the style of classical Russian schooling: disciplined exposition, careful definitions, worked examples, proof, and cumulative training.

That was exactly what made it valuable.

Replacing such books is not automatically progress. In mathematics education, “new” often means worse.

The child does not need a fashionable textbook. The child needs a clear path from counting to calculation, from arithmetic to algebra, from examples to proof.

Kiselev provided that path.

The reformers broke it.

Kolmogorov’s later role

Kolmogorov must be placed correctly in the story.

He was not some minor figure. He was one of the greatest mathematicians of the twentieth century, and his name became strongly associated with the later Soviet mathematics reform.

But that is precisely why the story is so important.

Being a great mathematician does not automatically make someone a great school textbook writer.

A school textbook is not a research monograph for children.
It is not a university course simplified for teenagers.
It is not a place for famous academics to impose abstract structures on pupils before they are ready.

A school textbook must understand the sequence of the child’s mind.

The later Soviet mathematics reform, often called the Kolmogorov reform, was part of a wider international “New Math” movement. Yury Neretin’s historical account explicitly says the reform involved grades 4–10 and names both Alexey Markushevich and Andrey Kolmogorov as leaders. It also says the program was too optimistic, insufficiently tested, and that from September 1972 Soviet schools were in a depressed state; after the reform was stopped by Lev Pontryagin in 1980, the system still did not return to its pre-reform level.

That is the real lesson.

The old books were not destroyed because they failed.

They were destroyed because reformers wanted to reform.

The false worship of modernisation

This is the disease of modern education everywhere.

A serious tradition is built slowly over decades. Teachers refine it. Textbooks improve. Children learn. Standards become clear.

Then the reformers arrive.

They announce that everything must be made modern. The old books are declared outdated. The old methods are called too strict. The old proofs are too hard. The old arithmetic is too formal. The old geometry is too demanding.

Then comes the new language: activity-based learning, discovery, relevance, modernisation, restructuring.

And after all this noise, what happens?

Children know less.

They cannot manipulate fractions.
They cannot reason through elementary algebra.
They cannot prove simple geometric facts.
They cannot calculate without a device.
They reach university with holes in the foundations.

This is not progress.

This is institutional vandalism.

Why I recommend these books

This is why I recommend old Russian textbooks.

Not because they are old.

Not because of nostalgia.

Not because everything Russian was automatically good.

I recommend them because many of them were built according to principles that modern textbook committees have forgotten.

They are concise.
They are cumulative.
They are serious.
They respect mathematics.
They respect the student.
They do not apologise for difficulty.
They do not pretend that understanding can be achieved without effort.

Kiselev, Rachinsky, Malinin and Burenin, and many others represent a lost standard of mathematical writing.

They show what a textbook can be when it is written to educate rather than entertain, to form the mind rather than satisfy a ministry, to teach mathematics rather than perform pedagogy.

The lesson

The sad story of Khrushchev-era reform, Markushevich’s educational bureaucracy, and the later Kolmogorov-branded mathematics reform is not only a Soviet story.

It is a warning.

A country can inherit a great intellectual tradition and still damage it.

A system can produce world-class scientists while simultaneously undermining the educational foundations that made them possible.

A bureaucracy can call destruction “modernisation.”

That is why old textbooks matter.

They are not museum pieces. They are evidence.

They show us that children can be taught seriously. They show us that clarity beats gimmicks. They show us that mathematics education once had a higher standard.

And they remind us of a painful truth:

Sometimes the best way forward is to recover what the reformers destroyed.

You can find these books here https://russianmathbooks.com/

A folk algorithm that turns out to be Egyptian, and turns out to be binary.

I spent some time translating Yakov Perelman’s Entertaining Arithmetic into English for the first time. Perelman was, in his lifetime, the most-read popular-science writer in the Russian language — forty books, thirteen million copies, translated into dozens of languages. He was the Martin Gardner of the Soviet century. In English, almost none of him exists.

One of the passages that stopped me during translation is a short section in Chapter 3, “The Russian Method of Multiplication.” Perelman quotes an 18th-century Russian arithmetic textbook — Magnitsky’s — which insists in verse that you cannot multiply without memorising the multiplication table. Then, quietly, Perelman notes that Magnitsky was wrong. There is a method for multiplying two numbers that requires no multiplication table at all. Only halving, doubling, and adding. Russian peasants used it. They inherited it, as Perelman puts it, “from deep antiquity.”

Here is the method. Suppose you want to multiply 32 × 13. Write the two numbers side by side. Halve the left number; double the right. Repeat until the left number reaches 1.

32 × 13
16 × 26
 8 × 52
 4 × 104
 2 × 208
 1 × 416

The answer is the last number on the right: 416. Verify: 32 × 13 = 416. ✓

The reason this works is disarming. Halving one factor and doubling the other preserves the product. So after six operations, we’ve reduced 32 × 13 to the equivalent 1 × 416 — which requires no multiplication at all.

What happens when the left number is odd? Halve after dropping the remainder — but on those rows, add the right-hand number back at the end. Example: 19 × 17.

19 × 17    (odd — keep 17)
 9 × 34    (odd — keep 34)
 4 × 68    (even — cross out)
 2 × 136   (even — cross out)
 1 × 272   (odd — keep 272)

Sum the kept rows: 17 + 34 + 272 = 323. And 19 × 17 = 323. ✓

Two things about this method are worth pausing on.

One: it isn’t Russian. Perelman traces it, in the following section, to Ancient Egypt. Scribes on the Rhind Mathematical Papyrus, around 1550 BC, were multiplying by exactly this method — successive doubling, tallying the powers of two that sum to the multiplier. The “Russian peasant” name comes from a British journal called Knowledge, which christened it that way just before the First World War. The method had migrated from the pharaohs’ land, along some unrecorded route, into the Great Russian village.

Two: it’s binary. Look at the odd/even column when you multiply 19 × 17. The odd rows are 19, 9, 1. In binary, 19 = 10011. Reading the odd/even pattern from the top: 1 (odd), 0 (even), 0 (even), 1 (odd — wait, that’s the crossed-out row) — no, let me realign. Halving 19 gives 9 remainder 1; halving 9 gives 4 remainder 1; halving 4 gives 2 remainder 0; halving 2 gives 1 remainder 0; halving 1 gives 0 remainder 1. Reading those remainders bottom-to-top: 10011 = 19. The peasant method is binary multiplication. It’s what every arithmetic-logic unit inside every modern computer chip does — shift left (double), shift right (halve), add when the bit is set. The Russian peasant and the Intel Core i9 are executing the same algorithm.

The reason I want to publish Perelman in English is that this is what he does on every page. Take a piece of folk knowledge that looks trivial. Unpack it, honestly. Show that it connects to something deep. Never sacrifice accuracy for showmanship. He did this for arithmetic, algebra, geometry, mechanics, astronomy — across forty books.

None of his mathematics books were in English until this month.

Entertaining Arithmetic — Yakov Perelman, translated by Valery Manokhin (Northern Star Academic Press, 2026). 151 pages, 66 problems with full solutions, 54 figures from the source edition.

Amazon (paperback + Kindle): https://www.amazon.com/dp/B0H8ZZLLGT
Gumroad (PDF): https://valeman.gumroad.com/l/perelman_arithmetic

~7 min read

TL;DR — In 2021 Amazon Science published “Probabilistic Forecasting: A Level-Set Approach,” introducing a tree method (LSF/XLSF, shipped in GluonTS as rotbaum) that reportedly “outperforms conformal predictions by a wide margin.” Two problems. First, its “conformal” baseline was a tabular regression method (CQR) on tabular datasets — not a forecasting conformal method at all. Second, when you run rotbaum through a proper, matched forecasting benchmark against a real conformal forecaster (CSP), it loses on the paper’s own headline metric, ties on CRPS, and produces 95% intervals that cover as little as 26%.

There’s a familiar template in machine-learning papers: invent a method, choose the comparison that flatters it, and announce a wide margin. The 2021 Amazon Science paper “Probabilistic Forecasting: A Level-Set Approach” is a textbook example — and it’s worth walking through, because the failure mode is common and the fix is simple.

The paper introduces the Level-Set Forecaster (LSF). Apply it to XGBoost and it becomes XLSF, and it ships in GluonTS under the module name rotbaum — German for “red tree.”The claim, in print: “XLSF outperforms conformal predictions by a wide margin on quantile prediction.”

That’s a strong, testable claim. So I tested it.

First: what is conformal prediction actually promising?

Quick primer, because it’s the crux. A probabilistic forecast doesn’t just give you a number — it gives you an interval, and that interval makes a promise. A 95% prediction intervalshould contain the true value about 95 times out of 100. That property is called calibration (or coverage), and it is the entire point of a prediction interval. Conformal prediction is the family of methods explicitly designed to keep that promise; most other models just hope.

Hold onto that word — coverage. It’s where this story ends.

The first problem: the paper never tested conformal prediction on forecasting

Look closely at what the paper actually compares. Its “conformal” baseline is CQR — conformalized quantile regression — evaluated on tabular, i.i.d. regression datasets: facebook, meps, concrete, bike, community. No forecasting conformal method anywhere in the comparison.

Benchmarking a forecasting tree against a tabular regression procedure and declaring victory over “conformal prediction” is like beating a chess engine at checkers and claiming the world title.

Conformal prediction for time-series forecasting is its own field — its own methods, its own guarantees, its own literature. The paper didn’t beat it. It beat something else and used its name.

So I ran the comparison the paper skipped: a real forecasting conformal method, on real forecasting data.

The fair fight

The opponent is CSP (Conformal Seasonal Pools) — a training-free conformal forecaster: no learned parameters, no neural network, milliseconds on a laptop (paper · code).

I dropped rotbaum into the exact audited protocol behind the CSP benchmark: the six datasets Amazon’s own DeepNPTS forecaster was introduced on — electricity, exchange rate, solar, taxi, traffic, Wikipedia — with the same 10 series each, the same 380 rolling-origin windows, the same 200 samples, the same proper scoring rules. Rotbaum uses GluonTS’s out-of-the-box XLSF settings and the same context length DeepNPTS got. Nothing tilted toward CSP.

One guardrail against hand-waving: re-running CSP through this harness reproduced its published, audited numbers to within 0.2% on every metric across all 380 windows. Everyone is measured with the same ruler.

The part that matters: the coverage is fake

Back to that promise. Rotbaum breaks it — badly.

Empirical coverage of the nominal–95% interval, per dataset. Rotbaum (red) is below target almost everywhere and collapses to 0.26 on Wikipedia; CSP (green) hugs the 95% line.

Across the six datasets, rotbaum’s 95% intervals covered 74% on average, and its 90% intervals covered 64%. On Wikipedia traffic it announced 95% confidence and delivered 26% coverage — the truth landed outside its “95%” interval three times in four.

A “95%” interval that actually covers 26% isn’t a sharp forecast. It’s fake certainty — and in a capacity-planning or risk system, it’s worse than no interval at all, because someone will trust it.

CSP sits near the line where it belongs: 89% average coverage, no Wikipedia-style detonation. (Exchange rate is genuinely hard — a near-random-walk where all methods undercover — and CSP’s honest 0.50 there is right in the chart. No cherry-picking.)

And the “wide margin”? A coin flip

Ignore calibration entirely and just score accuracy, and the promised blowout still doesn’t show up:

On weighted quantile loss — the exact metric the paper leads with — conformal prediction wins. On CRPS it’s a statistical dead heat: rotbaum wins 49% of 380 windows, Wilcoxon p= 0.45. There is no wide margin. There is barely a margin.

Where the tree does win — because honesty is the point

Rotbaum isn’t useless. It genuinely wins two of six datasets — traffic and taxi — on both CRPS and quantile loss. These are high-frequency, strongly seasonal, data-rich series with clean hour-of-day and day-of-week structure: exactly where a gradient-boosted tree thrives. If your problem looks like that, XLSF is a fair tool.

But “good on two of six, badly miscalibrated on the rest, beaten on the headline metric, and needs training” is not “outperforms conformal prediction by a wide margin.” It’s a competitive method oversold with the wrong benchmark.

The lesson: report the floor

Point forecasters have a discipline called Forecast Value Added — before you celebrate a model, show it beats a naive baseline. Probabilistic forecasting skipped that step, and papers like this one fill the vacuum: impressive-sounding gains measured against an opponent chosen not to win.

The fix is boring and it works:

Whenever you claim to beat conformal prediction, benchmark against a conformal method built for your task — and report calibration next to accuracy.

Had this paper done that, “wide margin” would not have survived the coverage column. The full argument is in Report the Floor and in my book, Probabilistic Forecasting with Conformal Prediction.

pip install git+https://github.com/valeman/csp-forecaster.git