The Master of the Royal Mint: How Much Money did Isaac Newton Save Britain? (original) (raw)

Summary

From the extant statistical data, the paper reconstructs several episodes in the history of the Royal Mint during Isaac Newton’s tenure. We discuss four types of uncertainty that are embedded in the production of coins, extending Stephen Stigler’s work in several directions. The jury verdicts in trials of the pyx for 1696–1727 allow judgement on the impartiality of the jury at the trials. The verdicts, together with several remarks by Newton in his correspondence with the Treasury, allow us to estimate the standard deviation σ in weights of individual guineas coined before and during Newton’s Mastership. This parameter, in turn, permits us to estimate the amount of money that Newton saved Britain after he put a stop to the illegal practice by goldsmiths and bankers of culling heavy guineas from circulation and recoining them to their advantage; a conservative estimate of savings to the Crown is £41510, and possibly three times as much. The procedure by which Newton probably improved coinage gives historical insight on how important statistical notions—standard deviation and sampling—came to the forefront in practical matters: the former as a measure of variation of weights of coins, and the latter as a test of several coins to evaluate the quality of the entire population. Newton can be credited with the formal introduction of testing a small sample of coins, a pound in weight, in the trials of the pyx from 1707 onwards, effectively reducing the size of admissible error. Even Newton’s ‘cooling law’ could have been contrived for the purpose of reducing variation in the weight of coins during initial stages of the minting process. Three open questions are posed in the conclusion.

1 Introduction: four sources of uncertainty in coining at the Mint

During its 900-year history, the Tower of London hosted not only the Royal Prison but also the Royal Mint. The Royal Mint aimed at producing large numbers of coins with known properties. The extant data enable statistical probing in several directions.

There were four sources of uncertainty that every Master of the Mint encountered. The first two were the weight and fineness of coins, which depended on a multilevel process of coining which formed the ‘population of coins’. The other two resulted from two kinds of sampling. One or two coins were picked from the production line at each ‘journey’ (from journée, a day’s work) and stored in a special box called the pyx. At the trials of the pyx from 1707 on, a number of coins, a pound in weight, was sampled from the total of all the coins in the pyx for a separate test.

The trial of the pyx is an ancient procedure that was intended to check the quality of coins and, hence, the Master’s skills. A public trial was held as early as 1248, though only in 1279 did King Edward I issue a proclamation describing the procedure to be followed. Several coins were taken from every production run of coined silver or gold and put into a special boxwood chest, called the pyx, which was then locked by the three Mint officers (Master, Warden and Comptroller). Periodically, initially every 3 months and later every 2 or 3 years, usually when the pyx was full, all the coins were taken out of the pyx to be counted and weighed by a special jury, which in Newton’s era included the King or his representative and the Lords of Treasury. The goal was to determine the average deficiency of coins in weight. To check the gold content (‘fineness’), a handful of coins was assayed and tried against a special trial plate prepared by the Company of Goldsmiths. During Newton’s tenure at the Mint, three such plates were in operation, made in 1660, 1688 and 1707, each slightly different in gold content.

To classify the four kinds of uncertainty, one must examine the minute details of each procedure pertaining to coining. Stigler (1977) studied various historical sources and concluded that the picking of the coins could be considered as random from the second quarter of the 19th century on, though not necessarily before then, when individual coins might have been distinguishable. The period from 1696 to 1727, the subject of this paper, presents a challenge to a historian. The intimate details of the minting process in Newton’s era are buried in archives or scattered throughout the diaries of his contemporaries in financial circles or in government. The quantitative data can be sifted out of Newton’s statistical observations on the activities of the Royal Mint contained among his personal notes and letters. Moreover, the minting procedure itself must be scrutinized to find the distribution of the weights of coins. When we ascertain that it was a normal distribution and estimate its standard deviation σ, we can answer several interesting questions, like the one in the title, ‘How much money did Isaac Newton save Britain?’.

The question is not idle. Newton’s contemporaries spoke with dismay of

‘an Invention which is called Culling: picking out the lightest pieces and putting them off and melting down the heavier, and sending them back again unto the Mint’.

We shall examine Newton’s statement that he put ‘an end’ to this activity and ‘thereby saved some thousands of pounds to the government’.

This paper is organized as follows. In Section 2, we present the legal parameters with which a Master of the Mint worked: the margin or remedy in weight (the legally permitted deficiency in weight) and the remedy in fineness (the legally permitted deficiency in fineness), usually expressed per pound of coins. These parameters are related to the modern notion of standard deviation, which was unknown to Newton and his generation of scientists. In Section 3, we review the results of the trials of the pyx during Newton’s time at the Mint. In Section 4, we estimate the inequality of coins in weight and refer to the practice of culling. In Section 5, we discuss nuances of the minting procedure, arguing that the distribution was symmetrical. In Section 6, we find the thickness of the guinea and explain why the coins’ distribution in weight must be normal. In Sections 7 and 8, we suggest an ingenious way to estimate the standard deviation σ of the weight of an individual guinea before and during Newton’s Mastership. The first scenario, which is given in Section 7, was tested but rejected in favour of that in Section 8. This allows us in Section 9 to calculate how much money Newton saved the British Treasury. In Section 10, we conjecture on a reform in the coining process that Newton introduced in the Mint to diminish the variation of coins in weight. In Section 11 we show how testing the small samples of coins effectively reduced the legal margin and claim this was an anticipation of standard deviation. Three open questions are posed in Section 12.

We use ‘Britain’ in this paper, although before the 1707 Union with Scotland the country was properly called England, whereas afterwards it became Great Britain. Until September 1752, Britain used the Julian calendar in which the year began on March 25th; therefore, for all dates from January 1st till March 25th, it became customary to assign a double year (_Y_−1)/Y, where _Y_−1 is the year according to the old style and Y according to the new style. We solely use the new style both for brevity and since the Mint’s financial year during Newton’s tenure began on January 1st.

The 17th–18th-century British system of legal weights used 1 lb troy = 12 oz; 1 oz = 20 dwt (pennyweight); 1 dwt = 24 grains, where 1 grain is equal to today’s 0.0648 g; thus 1 1b troy (or ‘apothecaries pound’) is 5760 grains or 373.24 g (not to be confused with an avoirdupois pound of 7000 grains or 453.59 g). Additionally, the weight was sometimes specified in carats, with 1 carat = 10 dwt = 240 grains. In the British monetary system of the time, £1 = 20s, and 1s = 12d. Both ‘pounds’—by weight and by value—were related mathematically: 1 pound troy (lb) in gold was divided into 44.5 guineas, 129.44 grains each (8.4 g). During Newton’s time, official documents often referred to a guinea as a ‘pound piece’ or a ‘twenty-shilling piece’, e.g. in the Mint reports or in the jury verdicts. However, at the turn of the 17th century, the guinea’s market price was higher than 20s: 22s from March 1696, 2112s from February 1699 and 21s after December 1717. As a consequence, a grain of gold was valued at 2.04d, 2d and 1.95d respectively, in three consecutive periods.

The data that are analysed in the paper can be obtained from http://www.blackwellpublishing.com/rss

2 Master’s indenture: two remedies

Newton entered the Royal Mint in April 1696 as a Warden, largely to help Charles Montague push through the Great Silver Recoinage (1696–1699). However, the major hero of this story is gold currency, chiefly the guinea. Silver was coined too rarely during Newton’s Mastership (1700–1727) to be treated here on a par with gold. The gold was represented by guineas and the guinea’s three derivatives: the ‘five-pound piece’ (5-guinea piece), ‘two-pound piece’ (2-guinea piece) and ‘ten-shilling piece’ (half-guinea). Their gold content (fineness) was 22 carats out of 24 carats with a remaining 12th part an alloy that initially consisted solely of silver. At some point copper began to be added to the alloy, probably to harden the coins.

All three of Newton’s Master’s indentures specify a total allowed variation in weight and fineness (‘remedies at the shear and assay’) as 2 dwt (48 grains) per pound of silver coins and a sixth of a carat (40 grains) per pound of gold coins (Westfall (1980), page 608). These remedies remained unchanged for the period from 1549 to 1815 (Challis (1992), pages 726–758), which includes Newton’s entire lifetime. Commenting on the notion of remedy, Stigler (1977), page 494, suggested that being ‘outside the remedy’ meant inevitable punishment of the Master, whereas being ‘within the remedy’ at first meant reimbursing the Crown for deficiency, but later (after 1550) came to mean ‘tolerance’ limits. In Newton’s era, punishment for being outside the remedy meant having the Master’s indenture revoked. There are only two recorded instances when the remedy was not met; both cases occurred before 1550.

The fact that the King could be present at the trials on par with public representatives testifies that the remedies were two sided: the King’s concern lay with the upper margin, whereas the public was concerned with the lower.

The share of each—the remedy in weight (the ‘remedy at the shear’) and fineness (the ‘remedy at the assay’), within 40 grains for a pound of gold and 48 grains for a pound of silver coins—is unknown. In fact, this provision was not clear even to insiders! In January 1708, Patrick Scott, a Deputy Master at the Edinburgh Mint, wrote two rather lengthy letters to Newton soliciting explanations (Scott (1967), pages 510–512). Newton’s answers are not preserved, though in the second letter Scott mentions Newton’s response to his first letter. Newton confirmed that both remedies were two sided and seemed to maintain (according to Scott) that remedies in weight and fineness were equal, at least for coining silver; otherwise, the coins had to be remelted and reminted. If the same reasoning held true for gold, the remedies had to be of 20 grains each.

Newman (1975), page 91, discussing the quality of the gold trial plates and gold coins, reported that the ‘remedy was 3.5 parts per 1,000’. The remedy seems to be the remedy in fineness. Since a guinea weighed 129.44 grains, 3.5 parts per thousand would mean 0.45 grains of gold per guinea, or 20 grains per pound, as derived above. Newman, however, uses anachronistic terminology: the ‘parts per 1,000’ came from a much later era, the 1870–1880s, from Chandler Roberts, rather than from archive research from Newton’s era.

Craig (1953), page 164, and Challis (1992), page 347, with reference to the Mint 1.4 papers (i.e. in the Record Books of the Royal Mint, in the National Archives, Kew), say that the Royal warrant of February 5th, 1663, specified the margins for the individual coins, 12 grain for a ‘one-pound piece’ (a future guinea) and the proportional margins for the derivatives. If true, the provision was overoptimistic, for such a restriction would have been impossible to achieve at that time. To be practical, the legal restriction on an individual coin must be 2–3 times greater than the standard deviation of individual coins, which for the guinea was about 1.3 grains as late as 1699, as we shall see below. Stigler (1977), page 500, also perceived the difficulty, saying

‘the Royal warrant of 1663 provided for limitation on the variation of individual coins, but there is no evidence that this was incorporated into the trial of the Pyx’.

Therefore the supposed 12-grain deficiency restriction on the guinea could only mean that an average coin within a pound of gold coins must not lack 12 grain. Though probably perceived by outsiders as another way to state that a pound of gold coins must not lack 2214 grains, this restriction is sensibly stronger than the latter by √44.5 and, effectively, is equal to a 313-grain margin for an individual guinea, which would have been perfectly reasonable for the standard deviation of 1.3 grains that is found below.

3 Trials of the pyx in 1696–1727

The consecutive trials in 1696–1727 show an increasingly tighter performance in coining (Table 1).

Table 1

Data from the jury verdicts at the trials of the pyx, 1696–1727 (Mint 7.130, following feature 50–80; from the National Archives at Kew)†

†

The figures in roman are taken directly from the verdicts; those in italics are computed.

Table 1

Data from the jury verdicts at the trials of the pyx, 1696–1727 (Mint 7.130, following feature 50–80; from the National Archives at Kew)†

†

The figures in roman are taken directly from the verdicts; those in italics are computed.

In Table 1 the figures in the third, fourth, fifth and seventh columns are taken directly from the verdicts, except for those in the fourth column for the 1696–1701 trials that were computed by factoring the third column by 44.5. The figures in the sixth and eighth columns are computed by subtraction. The figures in the seventh column can be also found in Ruding (1840), volume II, pages 461–462. The figures in the seventh column for the 1696 and 1697 trials are unavailable; that from the 1699 trial was found among Newton’s personal notes, preserved in Scott (1967), page 313.

The August 1701 trial warrants digression. Westfall (1980), page 608, states that the jury found the gold coins to be deficient by ‘a twentieth of a carat’ (12 grains) per pound. This certainly is a mistake—the official record, Mint 7.130 (feature 58), displays nothing of the sort. In fact, the 1701 trial was unique: it included only the content of the pyx (4212 guineas in total) from the last 5 months of Thomas Neale’s Mastership (August–December 1699) but neglected the portion that was minted in 1700 and the first half of 1701 under Newton’s Mastership. Newton signed his indenture only on December 23rd, 1700, a year after Neale’s death. Could the standard rule of depositing coins in the pyx have been suspended during this period?

The standard rule prescribed that a coin be taken and deposited in the pyx from each ‘journey weight’, which, since 1601, was defined for gold as 15 lb troy (Craig (1953), page 403). And, since from April 1663 on, 1 lb troy of gold was divided into 44.5 guineas, the rule became equivalent to depositing one coin, a guinea or a derivative, out of 667 guineas minted.

The annual figures for gold production in 1700 and 1701 are given by Craig (1953), page 416, and Challis (1992), page 691. (For some unclear reason, Craig translated the figures into pounds sterling for the guinea’s rate at 21s, though from February 1699 till December 1717 the guinea was rated at 2112s. See the explanatory note by Challis (1992) on page 341. In turn, Challis equated the guinea with the pound sterling, probably following the original definition of the guinea as a ‘one-pound piece’. As a rule, the Mint reported these figures in pounds troy.)

Of 120212 guineas produced in 1700, 180 coins had to be deposited in the pyx. Of 1190019 guineas produced in 1701, 1784 coins were to be deposited in the pyx. Since we cannot find them at the August 1701 trial, a logical step would be to check the next trial. At the July 1707 trial, the pyx contained 851 guineas. Out of this number, about 300 coins were placed in the pyx from January 1702 to mid-1707, as about 4500 lb of troy gold was coined at that period (Li (1963), page 155). So the period from August to December 1701 may account for 550 or even 700 guineas if those 300 coins were all half-guinea pieces. Assuming that in 1700 and 1701 the Mint minted mostly guinea pieces, the fate of at least 1264 guineas, which should have been deposited in the pyx after Neale’s death on December 23rd, 1699, until August 6th, 1701, is a puzzle. Even if in 1700 and 1701 only half-guinea pieces were minted, at least 564 pieces would be still missing.

Those 4212 guineas taken in August 1701 from the pyx were found to weigh 11 oz 8 dwt and 1 grain; a trivial rescaling for a pound shows a deficiency in weight of 29.45 grains, not 12 grains, as Westfall reported. This value, together with the 27-grain and 25-grain deficiencies in the 1696 and 1699 trials, all greater than a 20-grain or even a 2214-grain margin but accepted by the jury, illustrates that in practice only the combined deficiency in weight and fineness were counted.

4 Inequality of coins in weight and the practice of culling

In 1710, in a personal memorandum, Newton (Mint 19.2, and Hall and Tilling (1975), pages 86–87) provided information about the distribution of coin weights and the practice of culling:

‘When I came first to the Mint & for many years before, Importers were allowed almost all the Remedy & the money was coined unequally some pieces being two or three grains too heavy & others as much too light, & the heavy guineas were called Come-again-Guineas because they were culled out & brought back to ye Mint to be recoined (as was then the common opinion) & thereby the public Moneys called the Coynage Duty were squandered away to the profit of the Master & Moneyers & Goldsmiths and the new moneys, which remained after the heavy pieces were culled out & was put away by the Goldsmiths among the people, was without the Remedy. The money is now coined equally so that the culling trade is at an end.’

Take note of ‘two or three grains too heavy’ and ‘others as much too light’. The statement that gold money ‘which remained among the people’—after being culled by goldsmiths—‘was without the Remedy’ raises questions. To our knowledge, Newton did not check the coins in people’s pockets. Undoubtedly he referred to ‘A Third Letter to a Member of Parliament concerning the value of guineas’, which was found today among Newton’s papers with his notes in the margins (Jones, 1991), in which an anonymous author claimed that in 1699

‘the lighter Guineas of 5dwt 8gr or 5dwt 7gr are put off, and all the heavier guineas sent to the Tower after they past a pilgrimage thro the Melting Pot’.

His commentary, that the remaining money ‘was without the Remedy’, shows that the 1.44 grains, the difference between 5 dwt 8 grains and a guinea’s standard weight, exceeded the ‘individual remedy’ of a coin.

In one place, Newton wrote (Craig (1946), page 37, and Westfall (1980), page 609) that, before he became the Master, ‘every eighth’—and, in another place, ‘every fourth’—guinea was culled, melted, and reminted. Later, Craig (1953), page 212, averaged these two fractions to arrive at ‘possibly a fifth of all gold coin’—not bothering to explain how he obtained this fraction. True, the arithmetic average of 14 and 18⁠, at which Craig undoubtedly aimed, is 116⁠, which is quite close to 15⁠. Gjertsen (1986), page 363, unwittingly cited Craig’s ‘fifth’ as a fact.

According to Newton’s testimony, he put culling ‘to an end’ by making coins more uniform in weight. He clearly recognized the large variation in weight as the source of the problem. By finding the variation before and during Newton’s Mastership, we can discover how much money he saved the Treasury.

Referring to A Brief Memoir relating to the Silver and Gold Coin of Kingdom by Hopton Haynes, Newton’s weigher and teller, composed around 1702, Craig (1946), page 38, says

‘[Newton] insisted that the individual coins should be struck very nearly to their exact weight; for this purpose he laid down limits of individual variation a little wider than those to which the average had to conform, and proportionally more for small coins than large. As a grain of gold was worth 2d, culling importers lost an average profit of 212d to 5d per ounce of gold coins struck for them, and the average weight of coins entering circulation was raised in the same measure.’

Craig did not say what the ‘average’ and ‘individual’ variations were. Nor did he explain his figures, which equal 30–60d per pound of gold coins. Westfall (1980), page 592, writes that A Brief Memoir is ‘a panegyric to Newton’ that ‘drew heavily on material found among Newton’s papers, which he must have furnished’. Unfortunately, A Brief Memoir is unavailable to us. Regardless, whether by Haynes or by Newton, the above figures were certainly derived from the observation that the deficiency in weight of a pound of gold coins in the pre-Newton era ranged from 18 to 30 grains of gold (see the data in Table 1 for 1696–1701), which Newton later eliminated completely (at least, according to his own testimony).

Besides believing that public gain was equal (‘raised in the same measure’) to the cullers’ losses, Craig seems to have confused gross and net profit. As we shall show, these differed by a factor of 2, and only the lower figure cited by Craig is realistic.

5 Minting procedure in the Royal Mint at the end of the 17th century

The minting process at the end of the 17th century consisted of 10 steps as described by Samuel Pepys in his Diary for May 19th, 1663 (Wheatley (1952), page 123), after visiting the Royal Mint on that day as a guest of Henry Slingsby, a Master-Worker. His testimony is unique in fact and precision, as attested by Sir W. Chandler Roberts FRS and Chemist of the Mint in 1870–1882, who called Pepys’s description ‘interesting and singularly accurate’ (Chandler Roberts, 1884). The 10 steps were 1, assaying the bullion, 2, melting and casting the mould into plate (or bar; a bar was 22 in long, 1.5 in wide and 12 in thick), 3, drawing the plate (or bar) between two rollers, 4, a second drawing between finer rollers producing a fillet (or strip), 5, cutting out round blanks (about 1 in in diameter) from the fillet, 6, filing (or sizing) blanks, 7, annealing and straightening the blanks, 8, blanching the blanks (removing spots with acid), 9, edging the blanks, and 10, milling (or minting per se)—striking the blanks between two dies.

We are interested in how each step affected the shape of the distribution of coins dividing them into two groups. Filing, edging and milling targeted individual coins, whereas the other seven steps were applied to a number of coins. Therefore, the latter group may shift the mean whereas the first three steps affected the shape of the distribution. Newton described some of the effects quantitatively, though, unfortunately for our purposes, mostly for silver.

The melting was responsible for the coin’s fineness; the casting and two drawings for its thickness; the cutting, for its diameter. The guinea was about 1 in in diameter, though diameters of guineas minted under different kings were slightly different (Ruding (1840), volume III, pages 337–343, plates XXXIV–LX). Still the variation in diameter of coins must be dismissed as a chief reason for variation in their weight: the wrong diameter would imply a constant bias, upward or downward, in the weight of coins. This would be noticed rather quickly, immediately implicating the cutting machine, which could be easily adjusted.

Let us find the thickness of a guinea, h. Accepting 1 in =2.54 cm for the diameter of a guinea, 8.4 g for its weight and 18.5 g cm−3 for the density of the gold–silver–copper 11–12–12 alloy, the equation

π×1.272 cm2×h×18.5 g cm−3≡8.4 g

shows h = 0.9 mm. The guinea was quite a thin coin! To maintain uniformity of thickness between different coins ought to be quite a problem. After a bar passed the drawings, the fillet of 0.9 mm thickness had a total area of

22in×1.5in×1.27 cm/0.9 mm=466in2.

Thus a bar was expected to yield a maximum of 466 guinea pieces. This could have been indeed an exact number if the drawings increased the fillet’s width to exactly 2 in, though the major result of the drawings was lengthening the bar, more than 10 times in this example. Of course, the remainder of the fillet after the blanks had been cut out (known as scissell) was recycled to mint new coins. Therefore, the total yield from a bar could be greater than the first output by a factor of 4/π, i.e. about 593 guinea pieces.

The problem that was faced by the Master of the Mint regarding uniformity of guineas in weight becomes clear: how to maintain a standard thickness of 0.9 mm uniformly across the fillet. Two steps having a direct effect on the thickness of coins are casting and drawing of the plate. Pepys wrote in his Diary for May 19th, 1663 (Wheatley, 1952),

‘2. They [Goldsmiths] melt it [bullion] into long plates, which, if the mould do take ayre, then the plate is not of an equal heaviness in every part of it, as it often falls out. 3. They draw these plates between rollers to bring them to an even thickness all along and every plate of the same thickness, and it is very strange how the drawing it twice easily between the rollers will make it as hot as fire, ye cannot touch it.’

Here Pepys reported an important problem that was faced by the Mint workers throughout history. If the surface of the mould is not protected from oxygen, after the melting air bubbles occur in the centre of the plate (the bar). After drawing the bar between two rollers, these pores will lessen the density of the fillet, but the thinness may pass undetected. This is the first source for the variation in the weight of coins coming from the same bar. Additionally, drawing the bar through the rollers takes time: the reduction in thickness of the bars is accompanied by a slight increase in their width and a very great increase in their length, so it is generally necessary to cut partly rolled bars into two parts to keep them of convenient dimensions. Two horse-driven drawings of a single bar could take as long as an hour, during which the bar gradually cooled by 100–200 ∘C. The edge that passed the rollers first at a higher temperature was thinner than the other edge that had cooled. The second edge became colder and coarser, and therefore could not be reduced to the same thinness as the first. This certainly increased variation in the thickness of coins coming from the same bar. The deviations from standard thickness seem uncorrelated for subsequent bars. The conclusion is that the distribution of blanks in density and thickness arose from purely random errors in melting and casting and temperature imbalances in the bars during drawings.

The number of coins produced was large: about 730000 guineas during Newton’s Wardenship and about 11 million guineas during Newton’s Mastership. Normality of the distribution must follow from the central limit theorem, since the coins’ weights were uncorrelated and their standard deviation was bounded from above. Indeed, Newton claimed that he had not seen coins with an extra 4 grains above or below standard weight. Therefore the distribution of blanks in weight before filing ought to be a close approximation to a normal distribution.

However, simply projecting this reasoning onto the population of gold coins is problematic. On one hand, according to Table 1, the weight of gold coins on average was below standard, as much as 18–30 grains per pound in 1696–1699. On the other hand, Newton reported that the tails of the distribution were symmetrical (‘others as much too light’) but left open the question of what he considered as the distribution’s mean; either

• (a)
  the average weight or
• (b)
  the standard weight of the guinea.

In the second option, the central part of the distribution must be highly non-symmetrical. In principle, this could be a result of filing that targeted only the right-hand tail of the distribution, making the distribution asymmetrical. But how could filing result in an 18–30-grain loss? For that, the Mint workers had to file off almost the entire right-hand tail of the normal distribution. Indeed, the mean of the right-hand tail is √(2/π)σ = 0.8_σ_. Shifting the right-hand tail to the mean results in reducing the mean by 0.4_σ_. For example, for σ = 1.3 grains this amounts to 23 grains per pound, i.e. within the 18–30 grains range. However, in this scenario, fully half of the entire population of guineas would have been filed—an impossible feat! Besides, if the filing was done as described, then it is unclear how any guineas could ever ‘come-again’ and how Newton could ever have seen heavy coins with 2–3 extra grains.

If, however, Newton equated the mean with the average value, then the distribution was symmetrical, as it is, for example, in the normal distribution. To make this option viable, we first need to re-evaluate the status of the 18–30 grains.

6 How fair is Newton’s testimony?

Continuing the previous passage (‘When I came first to the Mint’), Newton offered the following testimony (Mint 19.2 and Hall and Tilling (1975), page 87):

‘While the Importers were allowed the advantage of the Remedy there wanted about 30 grains of fine gold in 4412 Guineas & about 34 grains of fine silver in 62 shillings. There is now the just quantity of Gold & Silver in the moneys, & there wants only about 15 grains of fine Copper in 4412 Guineas or the third part of a grain of copper in a Guinea wch want is of no value or consequence & is occasioned by the great fineness of ye trial pieces.’

Here Newton pointed to the fundamental difference between deficiencies of coins in weight before and after 1700. Before, a pound of coins missed about 30 grains of gold. After, only 15 grains of copper were missing though ‘of no value or consequence’.

The complaint about ‘the great fineness of the trial pieces’ proves that Newton wrote these lines soon after the August 1710 trial of the pyx, when the jury found that gold he coined was

‘in fineness by the Assay quarter of a grain worse than the standard in Her Majesty’s Treasury dated 25 June 1707’

(Mint 7.130). Newton devoted many pages (Mint 19.2) to justifying himself and to blaming the new 1707 trial plate for this misfortune. The figure ‘15 grains’ also confirms 1710 as the time of Newton’s testimony: in Table 1, the 15 grains can be found only as the average deficiency in weight at the 1710 trial.

The presence of copper in the guinea in 1710 is somewhat surprising. The Mint’s bylaws in that year allowed no addition of copper to a guinea of 22-carat purity, but certainly the Mint workers could have experimented with alloys on their own. As early as 1684 there was a ‘partisan plan’ by a Mint worker, to add copper when melting gold scissell—this could have gone unnoticed since the major metal from scissell was too pure; however, the attempt was thwarted (Challis (1992), page 356). Though the melting point of copper is higher than that of silver and gold, the Treasury could have been increasingly concerned with the coin’s wear, which copper would reduce. Challis (1992), page 486, writes (without citing a reference) that, during the 18th century, the alloy was ‘roughly half silver and half gold’ but does not say exactly when copper was first added to an alloy. Craig (1953), pages 103–104, writes that

‘the practice of the English Mint aimed at an alloy of three-fifths silver and two-fifths copper in the early 19th century, and probably throughout, for the proportion cannot be reversed without altering colour’.

Though the last remark sounds alarming, if Craig, a Deputy of the Master, vouchsafes that all guineas were of the same colour, then his conclusion must be correct. Extrapolating these testimonies back in time, we may assume that copper indeed was present in the guinea in quantity during Newton’s Mastership or even earlier.

There is, however, a problem with Newton’s testimony about the ‘want of 30 grains of fine gold in 4412 guineas’ in the pre-1700 era. Though it is difficult to judge what had happened ‘for many years before’, since in the 1692 trial the coins in the pyx were not counted but only weighted in sum, Table 1 provides some facts about the period when Newton ‘first came to the Mint’ as a Warden. For years 1696–1699, Table 1 gives deficiencies in weight in the range of 18–27 grains per pound with an average 2323 grains. Was it only of gold as Newton testified? If yes, why was the 20-grain remedy never applied? The lack of 25 or 27 grains of gold far exceeded the remedy in fineness (20 grains); however, it was never reported by the jury, whereas a comparatively minor 11-grain deficiency in fineness (⁠14 grain per guinea) was duly reported in the 1710 jury verdict.

Newton supported his former testimony with another from an ‘independent source’ (Mint 19.2; Craig (1946), page 38, misunderstood the passage):

‘Gold Importers made about £4 2d per ounce. The Goldsmiths now complain that their Gold doth not make £4 per ounce. It should make only £3 19s 834d. So much it hath made ever since the last trial of the Pyx.’

The difference between the two prices, £4 2d and £3 19s 834d per ounce, is equal to 63d per pound of gold. In the 1700s, with 2d for 1 grain of gold 63d were equivalent to 31.5 grains of gold. Note, however, that the second figure, £3 19s 834d, did not come from goldsmiths. Newton announced it on his own to support his estimate of a missing 30 grains of gold per pound of gold coins under Neale’s Mastership. However, Newton’s inference is faulty. His arithmetic proves only that Neale’s guinea was deficient by 30 grains per pound—but does not say whether it was of gold or of alloy.

Indeed, how could gold importers know how much they made? The ancient custom was that the Mint received gold by weight and paid back an equivalent weight in coin. Challis (1992), page 278, brings an example of its possible violation in 1617, but another case he cites shows that the practice could have been restored by 1622. If the money was lighter than standard, then, on redelivery, importers indeed received more coins than they had expected; however, they were not in a position to guess the fineness of the coins, which was ascertained only at the trials of the pyx. Therefore, a significant part of the deficiency in weight under Neale could have been alloy. This conclusion could explain why Neale’s coins were always inside the combined remedy and why he was never reprimanded at the trials of the pyx.

Actually, Newton himself indirectly supported this conclusion, claiming that the 1707 trial plate was ‘5/12 of a grain better than Standard’ and the 1688 plate was ‘a sixth part of a grain better than Standard’, whereas the 1660 plate ‘is Standard’ (Mint 19.2 and Hall and Tilling (1975), page 87). This is a much-quoted (Craig (1946), page 78, Craig (1953), page 216, and Newman (1975), page 94) but never properly understood passage. Since Newton’s coins (except for the 1710 trial) were tried against the 1688 trial plate, Newton actually believed that there was an extra 16 grain of gold in a guinea that he minted compared with a guinea that Neale minted. (This is despite the fact that Neale’s guineas were adjudicated against the same trial plate!) The difference amounts to 7.4 grains of gold per pound, which, together with Neale’s average deficiency in a guinea’s weight of 2323 grains per pound, suggests an estimate of 15–17 grains per pound for the loss in alloy under Neale, i.e. in the same range as Newton’s.

Is there any historic support for the alleged 7.4 grains? Craig (1953), page 217, cites the results of assaying guineas, after their withdrawal in 1773, from different reigns. The difference in a guinea’s average fineness between Newton (Queen Anne’s and King George I’s coins) and Neale (King William III’s coins) is 1.1–1.2 parts per thousand. Since 1000 parts ≡ 129.44 grains, this is equivalent to 6.3–7 gold grains per pound. So Newton’s claim was correct, though he never pressed this issue further, perhaps because he failed to discover its source. As a matter of fact, in the 18th century the fineness of guineas steadily, though almost imperceptibly, advanced: by the time of gold recoinage in 1773 it was greater than under Newton by another 1.1 parts per 1000 (Craig, 1953).

The source for the 7-grain difference should be sought in several Mint operations that affected the weight of the coins, resulting in its partial loss. A part of the loss could have come naturally from the melting process. The residual after melting, the so-called ‘waste’, was duly measured by Newton and ‘doth not amount to five grain in the pound weight of gold’ as Newton testified later in a 1709 letter to George Allardes, Master of the Edinburgh Mint (Scott (1967), page 536). Another part of the loss could have come from two consecutive operations over the blanks. Blanching led to their losing weight, whereas annealing led to their gaining weight. Summarily, this led to a loss of 4.5 grains per pound of silver, according to Newton’s research (Scott (1967), pages 256–258), but possibly per pound of gold, also. During his Mastership, Newton probably found a way to handle these operations better than Neale. It is important for us, however, that neither type of loss could change the shape of the (normal) distribution of coins in weight, although they could affect its average.

The third type of loss could have resulted from the operations applied to individual guineas: filing, edging and milling. Each contributed to a coin’s loss of weight, but filing was the major reducer. In his Diary for May 19th, 1663, Pepys (Wheatley, 1952) described filing as follows:

‘6. They [Moneyers] weigh these [blanks], and when they found any to be too heavy they file them, which they call sizeing them; or light, they lay them by, which is very seldom, but they are of a most exact weight, but however, in the melting, all parts by some accident not being close alike, now and then a difference will be, and, this filing being done, there shall not be any imaginable difference almost between the weight of forty of these against another forty chosen by chance out of their heaps’.

This is the important testimony that Stigler (1977) had been looking for but only found in later literature. At least in 1663, after filing, the coins were considered ‘close alike’ and were chosen for the trial of the pyx ‘by chance out of their heaps’.

Unlike other steps, including edging and milling, filing targeted only the right-hand tail of the distribution. Since Newton testified to having seen coins with 2–3 extra grains, perhaps, as Pepys observed, filing was done only for very large blanks, with 3 or more extra grains. A realistic scenario would be to size 5–6 blanks out of a bar or, say, one (the heaviest) blank out of 100. In the normal distribution, these blanks would be heavier than standard by 2.33_σ_ or more grains. If these blanks were sized to the mean weight, the average loss per coin would be

∫2.33σ∞zexp−z2/2σ2(2π)σdz=12πexp(5.43)σ=0.0265σ.

This is 1.17_σ_ for a pound of (44.5) coins. For example, for σ = 1.3 grains, as found below, this amounts to 1.5 grains. Actually, the loss of the far right-hand tail above 2.33_σ_ would not have a great impact on the symmetry of the distribution. If the ‘light’ coins were duly ‘laid by’, the far left-hand tail was also removed and the resulting distribution could be considered as fairly symmetrical, close to the normal distribution, though having both far tails cut off.

7 Standard deviation of guineas in weight: the first scenario

Let us return now to the practice of culling the heavy guineas during Newton’s tenure as Warden. To deal with the uncertainty regarding the fourth versus eighth ‘come-again-guineas’, let us average the two fractions for convenience. Taking the harmonic mean, we may assume that every sixth guinea ‘came again’ to the Mint. For a normally distributed population, the fraction 16 points to the coins with weight above 1 standard deviation, σ. But what was the value of σ at the end of the 17th century? Neither Craig, nor other historians suggest a figure. We shall deduce it indirectly.

As Warden, Newton was not involved personally in the production process; thus it is unclear when or where he had a chance to see coins with 2 or 3 extra grains. Therefore, two scenarios are possible for the circumstances under which he had seen such coins:

• (a)
  the minters showed all notoriously large coins to all the officers of the Mint or
• (b)
  Newton saw these unusually large coins only at the trials of the pyx.

Here we show that, according to the first scenario, the standard deviation σ for the weight of an individual guinea before 1700 was between 23 and 1 grain.

The proof comes from the following chain of observations. Newton claimed that, after his arrival at the Mint, he saw guinea coins weighing 2 or 3 extra grains. This is an important fact, especially coupled with an assumption of the normality of the coin distribution in weight. If σ was, for example, 34 of a grain, then a guinea of 2 or more extra grains was above 2.67_σ_ and therefore the a priori probability to produce such a coin was 4 in 103. A coin of 3 or more extra grains was above 4_σ_ and therefore its chances of appearing were 3 in 105.

But how many such guinea coins could Newton have personally seen between the time that he became Warden and becoming Master of the Mint, i.e. from April 1696 to December 1699? For this we have to know the amount of gold coinage in those years.

Challis (1992), page 342, gives annual figures of 120447 in 1697, 471567 in 1698 and 141377 guineas in 1699. In 1696, 138618 guineas were coined but, since it was enacted by Parliament that from March 2nd, 1696, till January 1st, 1697, the Mint was ‘under no obligation to receive or coin any gold whatever’ (Li (1963), page 124), the gold minting in 1696 must have taken place in January and February, i.e. before Newton assumed Warden’s duties. Thus, 733391 guineas were produced during Newton’s tenure as the Warden.

This number can be corroborated by an ‘inverse’ count. According to Table 1, the 1697, 1699 and 1701 pyxes contained 1039 guineas in total, which suggests that 693013 guineas were produced from August 1696 till December 1699. This is about 5% fewer than the reported number of guineas above. The reason for this discrepancy lies in the presence of the guinea’s derivatives. Surprisingly, the exact proportion of each derivative is unknown. Challis (1992), pages 431–432, provides rather meager information:

‘Five- and two-guinea pieces had never been struck in quantity.... Half-guineas were thinly scattered among the guineas....’

However, the near constancy of the content of the small samples in the trials of the pyx in 1699 and 1707–1727 suggests that guineas represented about 75% of the total amount of the gold coins. For example, Newton recorded (Scott (1967), page 313) that ‘1 five pound piece, 35 guineas and 9 half guineas amounting to 4412 guineas’ were sampled for a show at the 1699 trial. In most of the later trials, 33 ‘one-pound pieces’ were sampled among 44.5 guineas.

Accepting the 75% proportion, the number of guinea pieces minted during Newton’s tenure as Warden was about 550000. If the standard deviation of an individual guinea was, for example, 34 of a grain then, during Newton’s tenure as the Warden, the goldsmiths could have minted about 16.5 guineas (5.5 × 105 × 3/105) with 3 or more extra grains. In contrast, they could have coined about 2200 guineas (5.5 × 105 × 4/103) with 2 or more extra grains.

The true standard deviation might have been smaller, though it could not have been less than 23 of a grain, since in this case a guinea with 3 or more extra grains lies above 4.5_σ_, and, therefore, its chances of being coined are 3 out of 106. Hence the goldsmiths could have minted fewer than two such guineas (5.5 × 106=1.65⁠), a borderline case. Conversely, the standard deviation could not have been as high as 1 grain, since then Newton would have seen guineas with 4 extra grains. Indeed, such a coin lies beyond 4_σ_, and hence it would be possible to see about 16.5 (5.5 × 106 such coins—but Newton did not mention seeing any. Therefore 34 of a grain might be a good approximation for the standard deviation in this scenario. Let us see whether it withstands further scrutiny.

After entering circulation, the heavy guineas became prey for importers of gold (goldsmiths and bankers), who melted and returned them to the Mint for recoinage. Assuming that every sixth guinea came back to the Mint because they were oversized, for the normally distributed population, all ‘come-again-guineas’ were heavier than average by approximately 1 standard deviation. Could the cullers actually make a profit in such an environment?

The mean weight of those coins that are heavier than the mean by 1 σ is

∫σ∞zexp−z2/2σ2(2π)σdz=12πeσ=0.242σ.

This gives 10.77_σ_ grains per pound (44.5 guineas). If the 2.33_σ_-tail was filed off taking 1.17_σ_ grains per pound off, 9.6_σ_ grains remained. Since 1 grain of gold was worth 2.04d most of that time, the remainder was worth about 19.6_σ_d per pound.

Accepting σ=34 of a grain as above, the gross profit of cullers had to be about 14.7d per pound, quite an emolument! But now another obstacle arises: since the Mint would not accept guineas for reminting, the cullers had to melt them down into bullion before bringing them to the Mint. Indeed, the anonymous author of ‘A third letter to a Member of Parliament’ (Jones, 1991) describing culling, specifically mentioned ‘melting down the heavier [coins]’, before ‘sending them back again unto the Mint’.

The cost of private melting for goldsmiths themselves is unknown to us, so we refer to the Mint price, which for melting 1 lb troy of gold was 13d, as Newton testified in the same letter to Allardes (see above). This alone would reduce the gross profit of cullers to 1.7d, which was too little to be worthwhile. Moreover, part of it would be further spent on delivery. According to Challis (1992), page 415, in the 1770s, porters received from the Bank of England a payment of 20s per 100 ingots. Since the ingot was the bar yielding about 593 coins, this is equivalent to about 0.2d for a pound. This arithmetic implies that σ ought to be much greater than 34 of a grain, perhaps above 1 grain. But such a change would render the first scenario impossible.

8 Standard deviation of guineas in weight: second scenario

The second scenario is that Newton could have observed the heavy guineas only at the trials of the pyx when the chest was emptied of coins in public and all three officers of the Mint were present. Newton, a Warden, could have seen three such chests before becoming the Master: at the 1696, 1697 and 1699 trials. This scenario may change the estimate of σ.

According to Table 1, the 1696, 1697 and 1699 pyxes contained 233112 guineas. Together with the 1701 pyx, which held guineas from the later part of 1699, they had 2374 guineas in total, which Newton could have possibly seen at these three trials. As we showed above, guineas per se constituted about 75% of all the gold coins in the chest and, hence, the number of guinea coins that Newton actually could have seen at those three trials was 1780.

This scenario would lead to a re-evaluation of our earlier figure for σ. It must lie between 1 and 1.4 grains. Indeed, the former hardly allows guineas with 3 or more extra grains (1780 × 1.35/103=2.4); the latter allows about four guineas with 4 or more extra grains (1780 × 2.2/103=3.9) but Newton did not mention seeing them!

The figure σ = 1.3 grains looks quite realistic for two reasons. First, σ<1.3 grains is impossible in our scenario because 2.33_σ_ would be less than 3 grains, whereas the tail beyond 2.33_σ_ was cut off after sizing heavy blanks; as a result, Newton would have been unable to see a guinea that had 3 extra grains or even near to it. For example σ = 1.2 grains would prevent him from seeing a guinea that had an extra 2.8 or more grains, contrary to what he claimed. On the other side, σ = 1.3 grains would allow Newton’s seeing guineas that had between 2.9 and 3 extra grains: more exactly, about 1780 ×3/103=5.3 such coins. Second, accepting the figure of 12 grain as the margin for an average guinea per pound of coins, as stipulated by Royal warrant of February 5th, 1663, the effective margin for an individual guinea was 12 13 grains, or about 2.5_σ_, which guaranteed that a reasonably careful Master would never exceed it.

Accepting σ = 1.3 grains in the pre-Newtonian era, the previous estimate of 19.6_σ_d shows that before February 1699 the cullers robbed the English public of 2512d per pound. This was the gross profit for cullers. The net profit was 1212d because of the necessity to melt coins for 13d per pound. The profit is small but statistically certain.

9 How much public money did Newton save?

In his letter of August 12th, 1719, to the Treasury (Hall and Tilling (1977), page 58), Newton claimed that he had

‘brought the sizing of gold & silver moneys to a much greater degree of exactness than ever was known before & thereby saved some thousands of pounds to the government’.

But was it a few or was it many thousands? We can estimate this amount.

Recall that the value of gold was changed on December 22nd, 1717, by the Royal proclamation, which lowered the value of a guinea from 2112s to 21s. If the parameter σ = 1.3 grains remained unchanged, the cullers could have robbed the public by 19.2_σ_d=25d per pound before December 22nd, 1717, or by 19.2σd per pound of gold coins after that date.

During Newton’s tenure as Master, leaving aside problematic figures for 1700 and 1701, the Royal Mint minted gold (Li (1963), page 161) in the following amounts: £3.6 million under Queen Anne (1702–1714) and £8.5 million under King George I (1714–1727); altogether £12.1 million. However, we need a more refined count, for periods before and after December 1717. According to Newton’s report to Parliament, ‘from 1 January 1701/2 to 20 November 1717 his Majesty’s Mint within the Tower of London’ coined 145001 lb troy of gold (Hall and Tilling (1976), page 423). For the rest of Newton’s tenure till his death on March 20th, 1727, a direct count (Craig (1953), page 416, and Challis (1992), page 692) shows 4468686 guineas for 1718–1726 and 73194 guineas for the first quarter of 1727. This gives 102065 lb troy for 1718–March 1727. Since the price of gold varied in the two periods, we must keep the two figures separate but, altogether, from January 1702 till March 1727, Newton coined 247066 lb troy of gold or about 11 million guineas.

By counting the contents of the pyx from the trial of 1707 to that of 1727, 15899 guineas in total, and factoring the latter number by 667, we obtain 10604633 guineas. This is equivalent to 238306 lb troy of gold. The difference from the previous count, 8760 lb troy, or about 4%, is due to the presence of the guinea’s derivatives that made about 25% of the gold coins. The derivatives, however, pose the following problem: although they obviously ‘came again’ to the Mint with the cullers’ help, we have no clue how to measure their variability in weight since Newton did not comment on them. We have no choice but to deduce the variability through general reasoning. Since the intended margins (in 1663) for the derivatives were proportional to their respective values and therefore their weights, we may assume that their standard deviations were proportional to their weights also. Since a culler’s gain, in turn, was directly proportional to σ, we can substitute the derivatives by guineas alone.

By preventing cullers from taking advantage of the British public, Newton saved the public during his Mastership about 145001 × 25d/240=£15100 (from January 1702 till December 1717) and 102065 13£10350 (from December 1717 till March 1727), altogether £25450 of market money. Yet this money would not materialize until the first gold recoinage of 1773–1775. Had coinage remained the same as in the pre-Newton era, the gold recoinage either would have been more costly to Britain or would have occurred earlier.

However, putting an end to culling had another, immediate, side effect: after the cullers stopped circulating heavy money back to the Mint as bullion, Newton saved the British treas ury even more money—the Master’s share in Mint production. The payment to the Master’s team for coining 1 lb troy of gold was 6s 6d (Craig (1953), page 199).

Though we are not informed of the amount of additional circulation for the guinea’s derivatives, we may assume that it was the same as for guineas, i.e. every sixth coin. We may now estimate the projected amount of the ‘come-again guineas’ in circulation. Though every sixth coin would have been seized by cullers and returned to the Mint for recoining, this action could have been repeated as often as possible on every return. If repeated very often, the extra coinage could have been as large as a fifth of the total amount minted: 49413 lb troy. In this case, Newton saved an additional £16060 of the Treasury’s direct expenditures, if culling had continued at the same pace throughout the 27 years. Since, of 6s 6d, the Master’s personal gain was 22d, Newton conscientiously refused the extra bonus of £4530.

To sum up, Newton saved the Treasury £41510 in total. Given that the Master’s basic salary was £500 per annum, Westfall (1980), page 606, computes Newton’s average salary during his 27-year tenure as Master, which included rewards for coining gold and silver, as £994 per annum, or slightly less than £27000 for his entire life. Thus Newton saved the Treasury more than Newton as the Master earned during 27 years of service. No doubt the practice of culling would have persisted indefinitely, as shown by Newton’s report of November 10th, 1725, to the Treasury on Portuguese gold coins in Ireland (Hall and Tilling (1977), page 340).

This figure, however, could be an underestimation. The four Masters after Newton, up to the guinea recoinage of 1773–1775, capitalized on Newton’s improvements. Therefore, Newton’s improvements were of lasting importance beyond his death in 1727 and until at least 1772. The amount of gold coins minted was £11662215 during the reign of George II (1727–1760) and £8819417 during the reign of George III up until the recoinage (1760–1772) (Li (1963), page 161). This amounted to £20.5 million altogether, which is about twice as much as during Newton’s tenure. Thus, posthumously, Newton could have saved the Crown additionally twice as much as he saved during his lifetime.

10 How did Newton improve σ or what was his practical solution?

As we have seen, to bring culling to an end, Newton had to reduce σ to somewhere below 34 of a grain. This somewhat ad hoc value is nicely corroborated by the data from Table 1. Nine trials of the pyx, from the 1707 trial to the one of 1727, give a representative sample to estimate the value of σ during Newton’s Mastership. For this period, the figures in the fourth and fifth columns are summed up to 15860 and 15899 guineas respectively. From here, the average deficiency in weight of a pound of guineas is found as 5760×39/15899 = 14.13 grains. Taking this to be the mean of the distribution of the deficiencies in the eighth column, the estimate for the stantand deviation in weight of a pound of guineas comes out as 5.2 grains. Given the pound weight contains 44.5 coins, the standard deviation of the weight of individual guineas is 5.2/√44.5=0.78 of a grain.

For σ=34⁠, since 15899×3/105=/pagination 0.5, Newton would hardly have seen a guinea with an extra 3 or more grains during his Mastership. Besides, with 12 of a grain for the margin of the average guinea, the figure σ=34 of a grain may satisfactorily explain Craig’s words (above) that ‘Newton laid down limits of individual variation a little wider than those to which the average had to conform’. But Craig certainly used the wrong verb: Newton did not decree but rather achieved those ‘limits’.

Nowhere, however, did Newton state how he achieved his goal of stopping the culling of heavy guineas. Newman (1975), page 92, suggested that

‘Newton insisted upon coins being struck as closely as possible to their prescribed or standard weight, a practice that has of course been retained to this day’,

but forgot to inform his readers what Newton actually did. We must discover this on our own.

As Pepys indicated, there are two obvious ways to bring coins closer to their standard weight:

• (a)
  by drawing the plate between two rollers, thus making the fillet and, subsequently, the coins, more uniform in thickness, and/or
• (b)
  by filing (sizing) the blanks with a diameter larger than prescribed.

Each way influences the distribution of blanks in weight differently: the first reduces the variability; the second skews the distribution to the left.

Filing was used extensively before, though not necessarily during, Newton’s Mastership. Speaking of the drawings, Pepys wrote that the first followed the second:

‘They bring it to another pair of rollers, which they call adjusting it, which bring it to a greater exactness in its thickness than the first could be’.

Non-homogeneity of the plate in thickness led to non-uniformity of the coins in weight. Reducing non-homogeneity of the plate, and thus non-uniformity of the coins, could have led directly to reducing variability. To accomplish this feat, Newton could have demanded that the coiners bring a plate to a third pair of rollers. But could the story be more complicated?

Another way to keep the plate uniform in thickness during drawings, even without an increase in the speed of drawing, was to prevent its quick cooling. In 1701 Newton studied the process of cooling of different metals (Scott (1967), pages 361–364), deriving Newton’s so-called ‘cooling law’. Though copper has a higher melting point, it also has a specific heat capacity that is lower than that of silver and gold; therefore it cools more slowly. The latter factor could have been the main reason for Newton’s experimenting with ‘fine copper’ in alloy. He performed the initial experiments with various substances as early as March 1693, at Cambridge (Ruffner, 1963; Simms, 2004) but the first opportunity for experimenting with our three metals and confirming the quantitative form of the law for high temperatures could have offered itself to Newton only after he became Master of the Royal Mint.

11 Anticipating the standard deviation: the era of testing small samples

The purpose of the trials of the pyx was to reveal theft on the Master’s part, not to check on his professional skill. Stigler (1977), page 500, noted

‘The fact that the check on variability was not introduced in these early tests is evidence that the economic effects of increased variance (e.g., the ‘‘come again guineas’’) were not so great as the effects of changes in the average and that the very idea of measuring dispersion was not yet a well-formed concept’.

This was certainly true in the pre-Newton era. Westfall (1980), page 610, was the first to notice that Newton proposed checking not only the whole content of the pyx, but ‘arbitrary samples’ as well. Indeed, after the many-centuries practice of weighing the whole pyx chest, filled with as many as 1000 coins, Newton began weighing a set as small as a pound of coins, 44.5 guineas. As the consecutive jury verdicts witness, this test became an integral part of the trial of the pyx from the 1707 trial on, though the first evidence for this test comes from Newton’s personal note after the August 1699 trial of the pyx (Scott (1967), page 313).

Most likely, Newton was not the only inventor of such a test since the August 1699 trial was carried out under the Mastership of Thomas Neale. However, Newton could have been one of the inventors, since Neale was nearly on his deathbed in August 1699 and minting was supervised by John Francis Fauquier, the Master’s deputy, Thomas Hall, the Chief Clerk, and possibly James Hoare, Comptroller. By changing the size of the sample from ‘many’ pounds to just one, Newton and his colleagues effectively reduced the margin, as the law of large numbers states, by the square root of that ‘many’.

This may suggest that Newton was intuitively aware that variation could be measured, though he was not yet aware of how. Interestingly, Stigler (1977), page 498, was aiming to prove the same thesis by analysing, not the trials of the pyx (to which his paper was dedicated), but Newton’s figures for the range of Royal reigns in the ancient kingdoms in the posthumously published Chronology of Ancient Kingdoms Amended (1728), which ‘occupied much of Newton’s time and attention during the last ten or fifteen years of his life’ (Westfall (1980), page 808), i.e. during the last half of his tenure at the Mint.

Importantly, as can be seen from comparing the sixth and eighth columns of Table 1, the two deficiencies—the average deficiency in weight over the total amount of gold coins in the pyx per pound, and the deficiency of a sample pound—were very close over the years of Newton’s tenure. Though Newton, the Master of the Royal Mint, lived in constant tension with the Goldsmith’s Guild, the latter performed their public duty faithfully.

12 Conclusion

We have brought life to several of Newton’s personal notes, resulting in the estimation of the standard deviation σ of the individual guineas in weight before 1700 as 1.3 grains. By improving minting technology, Newton succeeded in reducing σ to 0.75 of a grain. This achievement allowed him to put a stop to the illegal, albeit widespread at the end of the 17th century, practice by bankers and goldsmiths of culling the heavy guineas from circulation and reminting them at public expense. In this way Newton saved the Treasury £41510. Of this sum, £16060 were the immediate effect of not reminting heavy guineas during Newton’s Mastership (1700–1727), whereas another £25450 became ‘visible’ at the first gold recoinage in 1773–1775, which was done at a cost to the government of ‘perhaps £750,000 or more’ (Challis (1992), page 440).

Challis (1992) says that ‘a Royal proclamation of April 1776 limited circulation of guineas to those that were within 1.5 grain of standard weight’. If true, we may assume that

• (a)
  the margin was at least thrice as small, or less than 0.5 grain, and
• (b)
  σ was at least twice as small as the margin, i.e. less than 0.25 grain.

This is three times better than the margin that was achieved by Newton.

Three historic puzzles remain. One is the conspicuous absence of coins minted in 1700 and 1701 at the August 1701 trial of the pyx. It is known that, following Newton’s report to the Treasury, the Order in Council of February 5th, 1701, prescribed that the French louis d’or and the Spanish pistole were not to be valued at more than 17s (Craig (1946), page 43–44). According to Newton’s testimony of 1717, this brought such a vast quantity of these coins to the Mint that £1.4 million gold coins (i.e. 1.3 million guineas) were minted from them (Hall and Tilling (1976), page 418). Since we did not find their samples at the 1701 trial and few at the 1707 trial, they seem to have been coming to the Mint over an extended period of time. Did this fact escape historians and mislead Craig and Challis regarding the 1700 and 1701 coinage?

Another mystery is when the ‘small’ sampling (of a pound of coins) was introduced in the trials of the pyx and on whose initiative. Though the first evidence for this test comes from the August 1699 trial of the pyx, it could have been introduced several decades earlier. Recall that Samuel Pepys in 1663 mentioned ‘any forty [guineas] chosen by chance’. Certainly, this quantity is too close to a pound in weight to be purely accidental. Could it be that testing small samples had begun in the 1660s, perhaps by Neale’s predecessor, Henry Slingsby, though not yet as a formal procedure incorporated into the trials of the pyx?

The third puzzle is the way that Newton achieved reduction in the variation of fillet thickness. Newton’s law of cooling could have been discovered while searching for the alloy with the lowest specific heat. The problem with that is that the table Newton published anonymously in volume 22 of the Philosophical Transactions (1701) as the long list of metals in ‘Scala Graduum Caloris’ does not contain gold, silver or copper, but stops with iron, which has half the melting point of the other three. Was the research on a gold–silver–copper alloy a state secret?

Acknowledgements

Thanks are due to Yaniv Stern (Leeds, UK) and Eduardo Vila-Echague (Santiago, Chile) for transcribing the Mint 7.130 papers, Lilla Vekerdy and Kirsten van der Veen (both of Special Collections of the Smithsonian Institution libraries) for help with examining Newton manuscripts at the Smithsonian Institution, Joan Griffith (Annapolis) and Sarah Olesh (Vancouver) for editing the style, Aaron Shapiro (Moscow) for advice on gold metallurgy, Randall Rosenfeld (RASC, Toronto) for helpful references, Stephen Stigler (University of Chicago) for advice on an earlier version of the manuscript and Richard Lockhart (Simon Fraser University) for several specific suggestions.

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