## Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines; una cum appendice, de aliâ eâque præstantiore logarithmorum specie contenda. Quibus accessere propositiones ad triangula sphærica faciliore calculo resolvenda: Unà cum annotationibus aliquot doctissimi D. Henrici Briggii, in eas & memoratam appendicem. Edinburgh: Andrew Hart, 1619. [Bound with:] GREGORY, James. Vera circuli et hyperbolae quadratura, in propria sua proportionis specie, inventa, & demonstrata. Padua: Giacomo Cadorino, .

First edition, extremely rare, of this complement to Napier’s epoch-making Mirifici logorithmorum canonis descriptio (1614) – while the Descriptio gave the first ever table of logarithms, it was in the Constructio that Napier explained the method of their construction. It is here bound with the first edition of James Gregory’s first mathematical work, highly important in the pre-history of calculus, and if anything even rarer than the Napier. “Probably no work has ever influenced science as a whole, and mathematics in particular, so profoundly as this modest little book [the Descriptio]. It opened the way for the abolition, once and for all, of the infinitely laborious, nay, nightmarish, processes of long division and multiplication, of finding the power and the root of numbers” (Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times (1958), p. 402). “The ‘Mirifici logorithmorum canonis constructio’ is the most important of all of Napier’s works, presenting as it does in a most clear and simple way the original conception of logarithms. It is, however, so rare as to be very little known, many writers on the subject never having seen a copy” (Macdonald, p. xvii). “Historically, it is important to note that in the Constructio the decimal notation is used with ease and power practically for the first time” (Henderson, p. 253). The second work in this volume is by the brilliant Scottish mathematician James Gregory. “Of British mathematicians of the seventeenth century, Gregory was excelled only by Newton” (Gjertsen, p. 245). The Quadratura “contained an astonishing number of novel and fundamental concepts, precisely formulated: concepts such as convergence, functionality, algebraic and transcendental functions, classes of transcendency, the process of iteration, the inherent likeness between circular and hyperbolic functions and the existence of functions invariant over an infinite sequence of values of their arguments. Incidentally, he calculated π to thirteen places and was the first to give the number 2.3025850929940456, or loge10, for the zone of the hyperbola. This battery of ideas was directed with the sole aim of proving the transcendence of π and e, an investigation that was finally completed by Lindemann at the close of the nineteenth century” (Turnbull, p. 5). “Although [his] proof was defective and in consequence rapidly incurred a storm of criticism it is to Gregory’s credit that he was the first to formulate a proposition of this class” (Baron, p. 231). “Indeed, by his speculations Gregory opens a new realm of mathematics … It is surprising that he quotes three important problems solved today: the squaring of the circle, the impossibility of solving the general algebraic equation, and the impossibility of reducing the pure equation of the nth degree [xn – 1 = 0] to quadratic equations” (Turnbull, p. 495). No other copies of either the Constructio or the Quadratura on ABPC/RBH in the last half-century.

Provenance: Earls of Macclesfield (South Library bookplate on front paste-down and blind-stamp on title of Constructio), (Sotheby’s, 4 November 2004, lot 885, £7,800). Erwin Tomash (book label on front paste-down).

The basic idea of what logarithms were to achieve is straightforward: to replace the wearisome task of multiplying two numbers by the simpler task of adding together two other numbers. To each number there was to be associated another, which Napier called at first an ‘artificial number’ and later a ‘logarithm’ (a term which he coined from Greek words meaning something like ‘ratio-number’), with the property that from the sum of two such logarithms the result of multiplying the two original numbers could be recovered. An idea of this kind was known to the Greeks: take an arithmetic progression (in which there is a constant difference between successive terms) and a geometric progression (in which there is a constant ratio between successive terms); writing one progression next to the other, one sees that adding any two terms of the arithmetic progression corresponds to multiplying the corresponding terms of the geometric progression. In the Constructio, Napier uses this idea but expresses it in kinematical terms. Whiteside suggests that Napier may have derived the idea of using motion in his construction from the writings of William Heytesbury and Nicole Oresme.

Napier (1550-1617) imagines two points, P and L, each moving along its own straight line. P starts at a point P0 and moves towards a fixed point Z in such a way that its speed is proportional to the distance PZ still to go, while L starts at the same time at L0 and travels at constant speed – if the constant speed of L is 1 we can think of L0L as the time taken by P to travel from P0. Napier defines the time L0L to be the logarithm of the distance PZ, which we will denote by NapLog(PZ). If the distance PZ = x, and the factor of proportionality r, so that the speed at P is x/r, it is easy for us to show (using calculus) that

NapLog(x) = r ln(r/x),

where ‘ln’ is our natural logarithm with base e. This means that, apart from the factor r, Napier’s logarithms are the same as our logarithms, but with base 1/e. Note that NapLog(r) = 0, so that r is the distance P0Z. For us the logarithm of 1 is zero, but Napier chooses r = 107 (see below).

How did Napier calculate his logarithms? If Q is another point and QZ = y, where y is less than x, the moving point is slowing down as it travels from P to Q so the time taken to travel the distance PQ = x – y is greater than the time it would have taken if the point had travelled at the speed x/r it had at P and less than the time it would have taken if it had travelled at the speed y/r it had at Q. The actual time taken is NapLog(y) – NapLog(x), so

(r/x)(x – y) < NapLog(y) – NapLog(x) < (r/y)(x –y).

If the difference between x and y is very small, the two end values are almost equal so we can take the middle term to be their average:

NapLog(y) – NapLog(x) is nearly equal to ½(r/x + r/y)(x – y).

Thus, if NapLog(x) is known, and y is very close to x, then NapLog(y) can be found.

But what if x and y are far apart? Napier’s procedure is complex, so we shall (over)simplify. Napier constructs a geometric progression ran, where a is a fixed number very close to 1 (and less than 1), and n is a positive integer. Napier calculated the terms of this progression by hand. From his definition of logarithms, it follows that

NapLog(ran) = n NapLog(ra).

Since ra is very close to r and NapLog(r) is known (= 0), NapLog(ra) can be calculated from the previous formula, and hence so can NapLog(ran) for all n. Suppose now that x is a number that is very close to ran for some value of n. Then, by the preceding formula again, NapLog(x) can be taken to be

NapLog(ran) + ½(1/an + r/x)(ran – x).

If the ratio a is small enough, every number x less than r will be close to one of the terms ran so this enables Napier to complete his table of logarithms. However, in order to achieve greater accuracy, Napier uses a more complex procedure involving three nested geometric progressions (these are set out in his three ‘Proportional Tables’), but the essential idea is the same.

In the Descriptio, Napier does not actually tabulate NapLog(x) for various numbers x, but rather NapLog(r sin(α)) for various angles α. This was because his tables were intended to be used to solve problem in spherical trigonometry, the type of problem most often encountered by astronomers. Trigonometric tables had been produced throughout the sixteenth century, and to avoid fractions it was usual to tabulate 107sin(α) rather than sin(α) itself (the choice of the factor 107 goes back to Regiomontanus). This is why Napier took the proportionality factor r to be 107. But Napier was not satisfied with 7 significant figures and actually used four more, beyond the ‘decimal point’, in his ‘Proportionalia Tertiae Tabulae’ – this is one of the earliest occurrences of our decimal point symbol in print, and it helped to stabilise this notation in its now-familiar form.

The enthusiasm with which Napier’s logarithms were received makes it clear both that this was perceived as a novel invention and that it fulfilled a pressing need. Foremost among those who welcomed the invention was Henry Briggs (1561-1630), Professor of Geometry at Gresham College, who wrote to the biblical scholar James Ussher in 1615: ‘Naper, lord of Markinston, hath set my Head and Hands a Work with his new and admirable Logarithms. I hope to see him this Summer if it please God, for I never saw Book which pleased me better or made me more wonder.’ Briggs did indeed visit Napier in 1615, and again the next year. Briggs convinced Napier of the advantages of having a version of his logarithms for which the logarithm of 1 is zero, and the logarithm of 10 is 1, i.e., our standard base 10 logarithms. Napier summarized his discussions with Briggs in the first Appendix in the Constructio, ‘On the construction of another and better kind of logarithms, namely one in which the logarithm of unity is 0’; this is followed by some remarks of Briggs. The second Appendix is devoted to some new formulas in spherical trigonometry, now known as ‘Napier’s analogies’, again followed by Briggs’ comments.

The Constructio was completed, at least in part, before the Descriptio, but Napier wished to delay its publication until ‘he had ascertained the opinion and criticism’ (Macdonald) of the Descriptio. He died in 1617 and the task of publishing the Constructio had to be completed by his son Robert, with assistance from Briggs. This first edition seems to have been issued together with a reprint dated 1619 of the Descriptio. The Constructio had a letterpress title page as well as a woodcut title-page intended to be used as a general title for the two works together and has the wording: Mirifici logarithmorum canonis description … accesserunt opera posthuma … Mirifici ipsius canonis constructio. As with most copies (e.g., that in Cambridge University Library, Syn.6.61.23), this one does not contain the reissue of the Descriptio – “Many copies lack the first part” (ESTC). The Descriptio and Constructio were reissued together at Lyon in 1620. While the Descriptio was translated into English in 1616 and into other languages soon after it appeared, and was reprinted many times, the Constructio had to wait until 1889 before an English version was produced.

“James Gregory (1638-75) was born into a scholarly family with established connections with mathematical and philosophical work. He received a sound mathematical education and, at the age of 26, having already written and published his first work, the Optica promota, made contact with the Royal Society on his way to Italy, where he studied for four years (1664-8)” (Baron, pp. 228-9). The Quadratura “was published at Padua in 1667, where Gregory was studying mathematics for some time. His teachers at the university introduced him to the ideas of Cavalieri and Torricelli [but] the source from which he is getting his inspiration is quite unknown to us. On the other hand, we find here a singular mixture of far-reaching ideas, exact methods, incomplete deductions, and even false conclusions.

“According to the title of the work, the essential part deals with the quadrature of the circle and the hyperbola. Archimedes had included the circle between inscribed and circumscribed polygons, calculated the perimeter of the 2n-sided polygon from that of the n-sided polygon, and by this had found the approximate value for the circumference of the circle. Gregory transforms this method into an algebraic one, to a sort of calculus. But instead of calculating the perimeters Gregory calculates the areas, and this enables him to apply the method simultaneously to the sectors of the circle, ellipse and hyperbola” (Turnbull, pp. 468-9). “Through the skilful manipulation of inscribed and circumscribed polygons he was able to generate a double sequence (an, bn) for the [area of a] sector of an ellipse, circle or hyperbola … After laying down the beginnings of a theory of convergence for such double sequences … Gregory attempted to establish the impossibility of rationally squaring the circle, ellipse or hyperbola by showing that no finite linear combination of the terms (an, bn) of the above sequence could result in a rational function of (a0, b0)” (Baron, pp. 229-230).

“Gregory finds in one process the area of sectors for the circle and for the hyperbola. Therefore we have here, for the first time, the analytical connexion between circular and hyperbolic functions or between trigonometric and exponential functions. And this discovery was made without using the imaginary numbers … Gregory was not only the first to discover the analytical identity of the two, seemingly quite different, functions but also was quite aware of the importance of the phenomenon” (Turnbull, pp. 471-2).

“When James Gregory, while living in Italy, published in 1667 his work on the quadrature of the circle and the hyperbola, he sent one of the copies of the very limited edition to Christiaan Huygens, in Paris, who was an authority in the subject; in a polite and even flattering letter he declared himself anxious to hear the opinion on his discoveries, of so competent a critic. Huygens never answered the letter directly; in July 1668, however, he published a review of the work in the Journal des Sçavans, in which he acknowledged its importance and the subtlety of its demonstrations, but at the same time raised several objections against the most remarkable proposition, and claimed his priority as to some of the author’s result” (Turnbull, p. 479). Gregory published a mild response in the Philosophical Transactions, to which Huygens replied in the Journal des Sçavans, rejecting Gregory’s explanations and stating that the main proposition of the Quadratura should be considered to be unproved. Gregory’s response was to publish a vigorous attack on Huygens in his Exercitationes Geometricae (1668). This time Huygens did not reply, and the Royal Society declined to intervene, despite Gregory’s urging.

A second edition of the Quadratura was published at Padua in 1668, issued jointly with and at the same time as Gregory’s companion work Geometria pars universalis.

Bound after these two important mathematical works is the third English edition of Henri Estienne’s emblem book:

The art of making devises: treating of hieroglyphicks, symboles, emblemes, aenigma's, sentences, parables, reverses of medalls, armes, blazons, cimiers, cyphres and rebus. First written in French by Henry Estienne, Lord of Fossez, interpreter to the French King for the Latine and Greek Tongues: translated into English, and embelished with divers brasse figures, by T[homas]. B[lount]. of the Inner Temple, Gent. whereunto is added, a catalogue of coronet-devises, both on the kings and the Parliaments side, in the late warres. London: Printed for John Holden, 1650.

This is a translation of L'Art de faire les devises, où il est traicte des Hieroglyphes, symboles, emblemes, aenigmes, sentences, paraboles, revers de medailles …, first published at Paris in 1645. English translations followed in 1646 and 1648.

Macclesfield 885; ESTC S123220 (Napier); Wing E3552 (Estienne). Tomash & Williams N4, Add13, Add18 (this copy). Baron, The Origins of the Infinitesimal Calculus, 1969. Gjertsen, The Newton Handbook, 1986. Henderson, ‘The Methods of Construction of the Earliest Tables of Logarithms,’ The Mathematical Gazette 15 (1930), 250-256. Macdonald, The construction of the wonderful canon of logarithms, 1889. Turnbull, James Gregory Tercentenary Memorial Volume, 1939. For further details on the construction of Napier’s logarithms, see Hobson, John Napier and the Invention of Logarithms, 1614, 1914; and Whiteside, ‘And John Napier created logarithms...,’ BSHM Bulletin 29 (2014), 154-166.

Three works bound in one vol., small 4to. Constructio: pp. [2, woodcut title], 67. Quadratura: pp. 63, with one folding engraved plate. Printer’s historiated device on title, woodcut head- and tail-pieces, framed initials. Date of publication from p. 59. Devises: pp. [2, engraved frontispiece], [viii], 87, with 10 unnumbered leaves with images of coins inserted between pp. 42 & 43 (light damp-stain to first few leaves of Constructio, some cropping of headlines and catchwords, slightly affecting text in Quadratura, engraved title of Constructio folded in at fore-edge and slightly cropped at head). Manuscript table of contents on front free-endpaper. Mid-eighteenth-century half-calf and marbled boards, spine with red-lettering piece and floral gilt ornament in each panel, red edges. Preserved in a cloth folding box with black morocco spine label.

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Item #4638

Price: \$18,500.00