Synopsis palmariorum matheseos: or, A new introduction to the mathematics: Containing the principles of arithmetic & geometry demonstrated, in a short and easie method; with their application to the most useful parts thereof: as, resolving of equations, infinite series, making the logarithms; interest, simple and compound; the chief properties of the conic sections; mensuration of surfaces and solids; the fundamental precepts of perspective; trigonometry; the laws of motion apply’d to mechanic powers, gunnery, &c. Design’d for the benefit, and adapted to the capacities of beginners.

London: J. Matthews for Jeff. Wale, 1706.

First edition, rare in commerce, of Jones’s second published work which “attracted the attention of Newton and Halley. Although the book was designed essentially for beginners in mathematics, it contained a fairly comprehensive survey of contemporary developments, including the method of fluxions and the doctrine of series” (DSB VII, p. 162). It was also in this work that Jones introduced the use of the Greek letter π to denote the ratio of the circumference of a circle to its diameter (Smith, History of Mathematics II, p. 312 & DSB VII, p. 163) – probably as it is the Greek letter corresponding to the first letter of ‘periphery’. This usage was popularised by Euler who adopted it in 1737 and used it in his series of great textbooks. In his Synopsis Jones (1675-1749) was one of the first to discuss fluxions in English and his acquaintance with Newton – he was one of the few privileged to have access to Newton’s manuscripts – led to his editing Analysis per quantitatum series (1711), one of the key texts in the history of calculus, and to his appointment to the committee set up by the Royal Society to report on the dispute with Leibniz over priority for the invention of calculus. The Synopsis was intended “for the Use of some Friends, who had neither Leisure, Conveniency, nor, perhaps, Patience, to research into so many different Authors, and turn over so many tedious volumes, as is unavoidably required to make but a tolerable Progress in the Mathematics.” “It was an advanced textbook which covered arithmetic and algebra in the first part, and in the second (‘containing the principles of Geometry’) conic sections, plane and spherical trigonometry, mechanics, and optics. Only a few pages were strictly devoted to the calculus of fluxions. Jones presented Newton’s notation, gave the rules of differentiation of the elementary functions and applied the ‘inverse method’ to power series. More interesting was the section dealing with mechanics, where Jones treated some propositions of the Principia. It was probably this section that rendered Jones’ Synopsis so useful at the beginning of the century in a period in which Newton’s major work was hard to understand even for the best mathematicians” (Guicciardini, pp. 14-15). The symbol π is first used on p. 243, and again on p. 263. Jones gives a method using infinite series to calculate the value of calculated π correct to 100 decimal places. This was in contrast to earlier methods, used by Archimedes, Viète, Van Ceulen, and others, which determined an approximate value of the circumference of a circle by using trigonometry to find the perimeters of inscribed regular polygons with large numbers of sides. ESTC records 22 UK and 16 US locations. However, only four copies have appeared at auction in the last 65 years, of which one lacked the plates. The Macclesfield copy, certainly Jones’ own, sold for £10,800 at Sotheby’s in 2005. The Synopsis was described as “quite extraordinarily rare” by J.J. Björnståhl in 1781 (Briefe aus seinem ausländischen Reisen, quoted by Harrison, Library of Isaac Newton, 859). Newton’s copy of the Synopsis is at Trinity College, Cambridge.

Provenance: John White (1699-1769), English politician of Wallingwells, Nottinghamshire, who sat in the House of Commons from 1733 to 1768 (signature dated 13.5.14 and bookplate on front paste-down).

Jones’ Synopsis opens with a somewhat fulsome dedication by the author to William Lowndes, Secretary to the Treasury, presumably in hope of patronage. “There follows the preface in which the author outlines the contents of the book, containing these words:

‘The whole is performed with as much Perspicuity and Planess, as the Subject and our Limits wou’d admit; And tho’ we have not bin over nice in ranging the Particulars of this Treatise, yet we carefully observ’d the method used by the most Eminent Mathematicians, who in their Writings were pleased to condescend to the Capacities of Beginners, as more especially in the Arithmetical part, Vieta, Oughtred, Tacquet and Wallis’” (Chambers, p. 194)

In the first part of the book, Jones treats elementary arithmetic and algebra, including arithmetic and geometric progressions, quadratic equations, the greatest common divisor and least common multiple found using the Euclidean algorithm, logarithms, and binomial expansions, with applications to the calculation of interest. This part of the book is well organised, and is perhaps based upon his teaching.

“The second part of the book occupies slightly less than a third of the whole book, and is entitled ‘The Principles of Geometry.’ Whereas the first part is divided neatly into sections and chapters, the second part seems to have been written hurriedly at a different time and suffers from lack of editing in the way of headings, sections, etc. The content is much more condensed and advanced than the first part. In this, what is virtually the differential calculus is introduced kinematically somewhat suddenly as follows [p. 226]:

‘And by considering Quantities as generated by continual Motion, ’tis apparent, that in equal Spaces of Time, they will become greater or less proportionally as the Celerity of the Motion by which they are so generated is greater or less: Hence the Celerity of Motion is very properly called Fluxion, and the Quantity generated Fluent. Now these Fluxions of Quantities are in the First Ratio of their Nascent Arguments; and may be express’d by Finite Quantities proportional to them.

And as x (by moving uniformly) becomes x + 0, xn becomes

(x + 0)n = xn + nxn-1 x 0 + ½(nn – n)xn-2 x 00 + &c.

But the augments 0, & nxn-1 x 0 + ½(nn – n)xn-2 x 00 + &c. are as 1 and

nxn-1 x 0 + ½(nn – n)xn-2 x 00 + &c. as 1 & nxn-1.

Th F, x : F, xn :: 1 : nxn-1’” (Chambers, pp. 196-7).

This is Jones’ proof from first principles that the fluxion, or first derivative in our terminology, of xn is nxn-1. An example in two variables follows. Altogether, differentiation is covered in four pages, the final remark being on the determination of maxima and minima using the fluxional calculus (p. 230):

‘Where the extreme value, whether Greatest or Least, of a Quantity is required; since it is an Invariable by Sup, and its Flux. = 0; Therefore to determine it to an Extremum, Put the Equation into Fluxions, let the Fluxion of that Quantity = 0; then will all the Terms wherein ’tis found Vanish, and the Extremum be determined by the remaining ones.’

“Later, a little use is made of the dot notation in dealing with some of the properties of conics. The definitions are followed immediately by some pages on geometry and series expansions” (Chambers, p. 198).

Jones first introduces the symbol π on p. 243. He makes use of the series expansion, due to the great Scottish mathematician James Gregory (1638-75), of the series expansion for the inverse tangent:

arctan(t) = t – t3/3 + t5/5 – t7/7 + t9/9 – t11/11 + …

Taking t = 1 gives

π/4 = 1 – 1/3 + 1/5 = 1/7 +1/9 – 1/11 + …,

but from a computational point of view this series is worthless. In his first letter to Leibniz, the Epistola prior, Newton wrote on 24 October 1676:

“To find the length of the quadrant to twenty decimal places, it would require about 5000000000 terms of the series, for the calculation of which 1000 years would be required” (Correspondence of Isaac Newton 2, pp. 138-9).

Jones takes instead t = 1/√3, so that arctan(t) = π/6, and writes:

“6 x (t – t3/3 + t5/5. &c.) = ½ Periphery (π)”

He goes on to write (still on p. 243):

“Theref. the … Diameter to the Periphery, as 1,000, &c. to 3.141592653.5897932384.6264338327.9502884197.1693993751.0582097494.4592307816.4062862089.9862803482.5342117067.9+, True to above a 100 Places; as Computed by the Accurate and Ready Pen of the Truly Ingenious Mr. John Machin”

Although the inverse tangent series converges more rapidly with t = 1/√3 than with t = 1, 206 terms of the series would still have to be taken to attain 100 decimal place accuracy, which was surely beyond Machin’s (or anybody else’s) tenacity or accuracy.

However, on p. 263 Jones returns to the subject, writing:

“the Diameter to the Circumference [is] as 1 to

(16/5 – 4/239) – ½(16/53 – 4/2393) + (16/55 – 4/2395) – &c. = 3.14159. &c. = π.

This Series … I receiv’d from the Excellent Analyst, and my much esteemed Friend Mr. John Machin”

Machin’s formula is, in modern terms,

π/4 = 4 arctan(1/5) – arctan(1/239).

The formula is easily established using the formula for tan (x + y) in terms of tan x and tan y. Machin’s own proof, which is ingenious and more geometric, was given in Francis Maseres’ Dissertation on the use of the negative sign in algebra (1758). Even using this formula, achieving 100 decimal place accuracy would have required very considerable labour and accuracy. However, Machin (1680-1751) was a very able mathematician. He was described by Newton as the man who “understood Principia better than anyone,” and in recommending Machin for the Professorship of Astronomy at Gresham College, Newton wrote that Machin was “in Mathematicks … a great Master” (Gjertsen, p. 331). Machin wrote important works on lunar theory which were incorporated into the third edition of Principia (1726) and the first English edition (1729).

Copies of the Synopsis are found described as having 1, 2, 3 or 4 plates (or none at all). There are three figures, numbered 1 – 3, in addition to numerous unnumbered figures within the text. In our copy these numbered figures are each printed twice: once on a single page opposite p. 246, and again in larger format opposite pp. 246, 253 and 253 (the pages to which the diagrams refer). We suggest that there are three issues of the Synopsis: ‘issue A’, with the three figures printed in large format on a single sheet which is either folded or cut up with the figures inserted at the appropriate points in the text; ‘issue B’, which has the figures in smaller format printed on a single page of the same size as the main text; and a hybrid of A and B with the figures present in both forms. Jones’ own copy, and those of Honeyman and Newton, are examples of issue A (with the three figures separated in each case); those at the Bayerische Staatsbibliothek, ETH (Zürich) and Göttingen are issue B; and the Bodleian and James Watt (Sotheby’s 2003) copies (and ours) are the hybrid issue (in the Watt copy the large format figures were present on a single folded sheet which had not been cut up).

William Jones was born in 1674 on a small holding close to the county town of Llangefni in the middle of the island of Anglesey in Wales. When he was still a small child the family moved a few miles further north to the village of Llanbabo. He attended the charity school at nearby Llanfechell, where his early mathematical skills were drawn to the attention of the local squire and landowner, who arranged for Jones to go to London, where he was given a position as a merchant’s accountant. He later sailed to the West Indies, an experience that began his interest in navigation. When he reached the age of 20, Jones was appointed to a post on a warship to give lessons in mathematics to the crew. On the back of that experience, he published his first book [A new compendium of the whole art of practical navigation] in 1702 on the mathematics of navigation as a practical guide for sailing. On his return to Britain he began to teach mathematics in London, possibly starting by holding classes in coffee shops for a small fee.

As a peripatetic teacher in London, within a year or two Jones was engaged as tutor to Philip Yorke, the future lord chancellor and first earl of Hardwicke. Lodging with John Harris FRS, author of the Lexicon technicum (1704), Jones probably contributed the navigational articles in the volume. In 1706 his teaching notes were printed under the title Synopsis palmariorum matheseos, or, A New Introduction to the Mathematics … Maybe the book, with possibly a recommendation from Harris, brought Jones to the attention of Newton. At this point, aged about thirty, Jones had delineated the two themes of his life – tutor to prominent figures and disseminator of Newton’s writings …

“Early in 1709 Jones unsuccessfully applied for the mastership of Christ’s Hospital Mathematical School, with a testimonial from Halley, and one from Newton, who wrote as if he knew Jones only by repute. In 1706 Yorke had embarked on a legal career and presently developed a close relationship with Thomas Parker, later first earl of Macclesfield. Through this connection Jones entered the Parker household as tutor to the son, George.

“In 1708 Jones acquired the mathematical papers of John Collins. At that time, the priority dispute between Newton and Leibniz, publicly ignited by Fatio [de Duillier] in 1699, was still smouldering. While unequal to evaluating Newton’s work critically, Jones had a good command and appreciation of it; he was already contemplating an extensive edition when he found a Collins transcript of a Newton manuscript, ‘De analysi’ (1669). Using this and other transcripts, together with guidance from Newton and borrowed autographs, he published Analysis per quantitatum series, fluxiones ac differentias (1711) … In his preface Jones presented ‘powerful evidence of Newton’s mathematical originality as far back as 1665’ (Correspondence of Isaac Newton 5, p. 95). It was a key element in the escalating dispute.

“Jones was admitted FRS on 30 November 1711, and was included on a committee appointed on 6 March 1712 by the Royal Society to inspect the letters and papers relating to the dispute. The documents came largely from the Collins papers. Selection, annotation, and editing were performed by Newton himself, with Jones helping to take care of the impression. The resulting publication, ostensibly from an impartial standpoint, was the anonymous Commercium epistolicum (1712) …

“During Jones’s long association with the Parkers he spent much time at their castle at Shirburn. The epigram ‘Macclesfield was the making of Jones, and Jones the making of Macclesfield’ testifies to Jones’s services in resolving the family’s ‘disturbance’ over an ‘Italian marriage’ (Hutton) of George Parker, although there is another interpretation. Jones lost heavily when his banker failed, but was supported by sinecures of secretary of the peace, procured by Hardwicke, and deputy teller to the Exchequer, by George Parker … His collection of some 15,000 books was considered to be the most valuable mathematical library in England and was bequeathed to George Parker, the second earl of Macclesfield” (ODNB).

In 2001, that part of Wiliam Jones’s collection that comprised papers and notebooks belonging to Newton were sold to the library of the University of Cambridge. The bulk of the rest of the library was sold in a series of auctions at Sotheby’s in 2004 and 2005.

“Some mystery remains regarding the fate of William Jones’s personal papers. The Macclesfield family had been reluctant to release them and there is the suggestion of a scandal that the family has sought to conceal. Those papers would surely throw further light on William Jones, on his relationship with the earls of Macclesfield, and on his remarkable life-journey from a cottage in Anglesey to be a member of the mathematics establishment” (The Guardian, 14 March 2015).

ESTC T90337. Sotheran 9894. Chambers, ‘The tercentenary of π,’ The Mathematical Gazette 90 (2006), pp. 194-202. Gjertsen, Newton Handbook, 1986. Guicciardini, The Development of Newtonian Calculus in Britain 1700-1800, 1989.

8vo (187 x 111mm.), pp. [xii], 304 (p. 153 misnumbered 155), woodcut diagrams and headpiece, four engraved plates (small ink stain to outer margin of a few leaves). Contemporary calf, spine gilt in compartments, red morocco lettering-piece, covers with double gilt ruling (spine and corners carefully restored, front board rubbed).

Item #5313

Price: $4,850.00