[BAHĀʾ-AL**-**DĪN ʿĀMELĪ, SHAIKH MOḤAMMAD B. ḤOSAYN BAHĀʾĪ]. Khulāṣat al‐ḥisāb [The Summa of arithmetic *or* Quintessence of calculation]. [*Near or Middle East*], *1075 AH* [*1664 CE*].

**£14,500**

A very early and well preserved copy of this important and influential mathematical work by Shaykh Baha’ al-Din (1547-1621), famed for his excellent command of mathematics, architecture and geometry, and one of the earliest astronomers in the Islamic world to suggest the possibility of the earth’s movement prior to the spread of the Copernican theory.

BOSCOVICH, Ruggiero Giuseppe. De viribus vivis dissertatio . *Rome, Komarek for Monaldini, 1745.*

**£5000**

First edition of this rare work by the great Croatian scientist, and which contains the first published statement of Boscovich’s universal force law, the basis of his atomic theory of matter, and which prefigures much of his masterpiece *Philosophiae Naturalis Theoria redacta ad unicam legem virium in natura existentium* of 1758.

In spite of his important contributions to mathematics, astronomy, and optics, today Boscovich is best known for a unified theory of matter based on the hypothesis of point atoms governed by a universal law of force.

The basic features of this remarkable theory go back to a treatise of 1745, *De viribus vivis* (Kragh, p. 19).

The fundamental features of his theory on indivisible points and the unique law of forces were first published by Boscovich in 1745 in a dissertation entitled *De viribus vivis*. Further developments of his theory were published in *De continuitatis lege*, 1754, then in *De lege virium in natura existentium*, 1755 (Markovi, p. 129).

The ‘Theory of Natural Philosophy’ is now recognized as having exerted a fundamental influence on modern mathematical physics. As the title of his book implies, he considered that a single law was the basis of all natural phenomena and of the properties of matter; that the multiplicity of physical forces was only apparent and due to inadequate mathematical knowledge (PMM).

[EUCLID]. DECHASLES, Claude-François Milliet. The Elements of Euclid explain’d in a new but most easie method ... *Oxford, Lichfield for Anthony Stephens, 1685.*

**£2250**

A very attractive copy in a contemporary English binding of the first edition of this English translation of Dechasles’s *Euclidis Elementorum libri octo, *a paraphrase of Euclid’s *Elements*.

This work covers Books 1 to 6, together with Books 11 and 12, of Euclid’s *Elements.* Another, more common English edition was published in London by Phillip Lea in the same year, the translation being by Reeve Williams.

‘Dechales [is also known to have] adopted Galileo’s theory of motion, where he introduced several original views and developments.

FELICIANO, Francesco. Libro di arithmetica et geometria speculativa et praticale … intitulato Scala Grimaldelli ... *Venice, Francesco Bindoni and Maffeo Pasini, 1536.*

**£4850**

The scarce second edition of Feliciano’s most famous work (following the publication of his *Libro de abaco* of 1517), generally known as the *Scala Grimaldelli* (first, 1526, and extremely rare).

‘Feliciano’s second work was highly esteemed as a textbook for schools ... more complete than the Treviso book, more modern than Borghi, more condensed than Paciuolo, few books had greater influence on the subsequent teaching of elementary mathematics’ (Michele Cigola, editor, *Distinguished Figures in Descriptive Geometry and its Applications for Mechanism Science* pp. 53).

MALFATTI, Gianfrancesco. Della curva cassiniana e di una nuova proprieta’ meccanica … *Pavia, at the Monastery of S. Salvatore, [1781].*

**£2250**

First and only edition of Malfatti’s treatise on Cassini’s curve.

‘Malfatti became famous for his paper *De aequationibus quadrato-cubicis disquisitio analytica* (1770), in which, given an equation of the fifth degree, he constructed a resolvent of the equation of the sixth degree, that is, the well-known Malfatti resolvent. If the root is known, the complete resolution of the given equation may be deduced … [In the present work] Malfatti demonstrated that a special case of the Cassini’s curve, the lemniscate, has the property that a mass point moving on it under gravity goes along any arc of the curve in the same time as it traverses the subtending chord’ (DSB).

In 1799 Gauss was to discover the relationship between the lemniscate sine function and the arithmetic-geometric mean iteration, which pried open the entire field of nineteenth century elliptic and modular function theory.

Newton, Sir Isaac. The Method of Fluxions and infinite series with its application to the Geometry of Curve-Lines … To which is subjoin’d, a perpetual comment upon the whole Work … *London, Henry Woodfall, 1736*.

**£27,500**

First edition, an interesting copy with some early corrections or notes in ink and pencil, of Newton’s work on Fluxions, ‘one of his greatest mathematical works’ (Cambridge Companion to Newton).

‘In the *Method*, Newton gives the solution of a series of problems “in illustration of this analytical art,” mainly problems of maxima and minima, tangents, curvatures, areas, surfaces, volumes and arc lengths. With qualities represented as generated by continuous flow, all of these problems can be reduced to the following two (one the inverse of the other).

1. Given the length of the space at every time, to find the speed of motion at any proposed time.

2. Given the speed of motion at every time, to find the length of the space described in the proposed time.

This is among the greatest generalizations in the history of mathematics, reducing the great majority of problems faced by mathematicians of the time to two basic problems’ (*Cambridge Companion to Newton).*

This is Newton’s fullest exposition of the calculus; though the last of his works on calculus to be published, it was the work which he himself intended to publish first, in Latin, in 1671. The first page of the manuscript (preserved in Cambridge University Library) is lost and the title *De Methodus Fluxionum* was supplied by John Colson when he first published it in this translation, with his own extensive commentary.

[NEWTON]. [PFAUTZ, Christoph]. Isaaci Newton, Matheseos Professoris Cantabrigiensis, & Regiae Societatis Anglicanae Socii, Philosophiae Naturalis Principia Mathematica. Londoni, jussu Soc. Regiae, 1687, in 4. [*contained in*:] Acta Eruditorum Anno M DC LXXXVIII publicata, ac Serenissimo Principi Ac Domino Dn. Friderico, Regnorum Daniae ac Norvegiae Haeredi &c. &c. Dicata. *Leipzig, J. Grossius and J.F. Gleditsch, 1688.*

**£4750**

The highly important *Acta eruditorum* review of the *Principia*.

There were four early reviews of the *Principia*: the first appeared in no. 186 of the *Philosophical Transactions* but, ‘not only did Halley finance, edit, publish and distribute *Principia*, he also reviewed it, anonymously, in *P[hilosophical] T[ransactions]*. It is little more than a summery interspersed with expressions of praise’. The second appeared in the *Bibliothèque Universelle *of March 1688, consisting ‘of nothing more than the headings of the sections of Books I and II translated into French. There is also a summary of Book III, and an introductory paragraph …’ The final review was that in the *Journal des sçavans*, August, 1688, in which ‘Newton’s hypothesis was dismissed as arbitrary, unproven and belonging to geometry rather than mechanics’.

Published in June, 1688, the review in the *Acta* is the third in sequence, and ‘the most detailed and serious of the four reviews. It was comprehensive enough to provide many people in Europe without access to the *Principia* itself with a fairly full account of its contents’ (Gjertsen, *The Newton Handbook* p. 472).

ROCCO, Antonio. Esercitationi filosofiche di D. Antonio Rocco filosofo peripatetico. Le quali versano in considerare le positioni, & obiettioni, che si contengono nel Dialogo del Signor Galileo Galilei Linceo contro la dottrina d'Aristotile. Alla santita di N.S. Papa Urbano VIII. *Venice, Francesco Baba, 1633.*

**£21,500**

First edition of this important and rare critique of Galileo’s *Dialogo*, published within a year of the *Dialogo,* and the work to which, as a consequence, much of the Galileo’s *Discorsi e Dimostrazioni Mathematiche, intorno a due nuove scienze* (1638) was written as a reply.

Rocco’s *Esercitationi* prompted Galileo to explain ‘how he detected and corrected the falsehood in Aristotle’s law of free fall’ (Shea) and formulated his own law of falling bodies. Wallace, examining the reasons why the Aristotelians are accorded better treatment in the *Two new sciences*, as compared to that in the *Dialogo*, remarks that ‘a factor that is noteworthy was the publication of a book in late 1633 and dedicated to Pope Urban VIII that defended Aristotle’s teaching against the attacks made by Galileo in the *Dialogo*. The author of the work entitled *Esercitationi Filosofiche*, was Antonio Rocco, and it is to Galileo’s credit that he read and annotated Rocco’s critique and even wrote out a series of replies to him, some of which later appeared in the *Two new sciences*’.

SCHOOTEN, Frans van. De Organica Conicarum Sectionum in Plano Descriptione, Tractatus. Geometris, Opticis, Prasertime verò Gnomonicis Mechanicis utilis. Cui subnexa est Appendix, de Cubicarum Aequationum resolution. *Leyden, Elzevier, 1646*.

**£8500**

The de Thou copy of van Schooten’s work on conic sections, a work studied by Newton.

‘Schooten’s first independent work was a study of the Kinematic generation of conic sections (1646). In an appendix he treated the reduction of higher-order binomial irrationals to the form x + √y in cases where this is possible, using a development of a procedure of Stifel’s. An interesting problem that Schooten considered was how to construct a cyclic quadrilateral of given sides, one of which is to be the diameter - a problem that Newton later treated in the lectures on *Arithmetica universalis* (*Mathematical Papers*, V, 162–181).

‘Fascinated by the personality and ideas of Descartes, he worked hard to popularize the new mathematics; his highly successful efforts assured its triumph’ (DSB).