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The title of Chapter 7 promises more than it delivers, "Incipit septima distinctio de inuentione altitudinum rerum eleuatarum et profunditatum atque longitudinum planitierum." Regardless of promises, the chapter offers only trivial exercises in measuring heights of trees, to judge their suitability as masts for ships. Not a word about finding depths, say of wells. It is curious that some of the Italian translations do offer examples of measuring depths of wells. Could this be evidence of a defect in the exemplary text followed by the copyist of, say, Urbino 292?The focus of the chapter is on measuring heights, using similar triangles, as is done in most middle schools today, although in two different forms. The first is a quadrant, also called by him "an oroscope," for which he presented detailed instructions for both construction and use. The second is "a wooden triangle." It appears in [4] as a 3-4-5 right triangle for measuring heights, again by similar triangles. Despite the practicability of these instruments, the incompleteness of the chapter is bothersome: there is nothing about finding depths or longitude of planets. All the other chapters are quite complete; a few are even overflowing with information. Why does this chapter seem to have been clipped of some contents?The Latin manuscripts offer no suggestion. The Italian manuscripts, on the other hand, do offer problems seeking depths of wells. Yet none of the material in the corresponding chapter of the Riccardiana manuscript appears here. Furthermore, the figures do not always match the text, and the text was in need of severe correction, especially in [7]. New figures were drawn. Note the use of the word equiangular (corresponding angles are equal) that is so different from modern use (all angles are equal). Finally, there is a single example where the Pythagorean theorem is used. [ABSTRACT FROM AUTHOR]

The second longest chapter, Chapter 6 may be described as Fibonacci's contribution on solid geometry. It offers a plethora of guidelines for finding the various dimensions of all sort of three-dimensional bodies, with alternate methods for many cases not to overlook proofs for many assertions. After a set of definitions that identifies foci of interest, Fibonacci announced the purpose and tripartite division of the chapter, to measure the dimensions of parallelepipeds, pyramids, and spheres. Thereafter he named all the Euclidean propositions from Elements, Books XI through XV [sic], that he might use. From there on he subjected his reader to methods for finding the surface areas and volumes of these figures: parallelepipeds, wedges, columns, pyramids, cones, spheres, and other solids. Noteworthy is that he used both arithmetic and algebraic tools to find dimensions. Fibonacci concluded the exercise with a discussion about finding the volumes of solids inscribed in one sphere, whereby knowing the volume of one inscribed solid the volume of another inscribed in the same or another sphere could be found. A few details about the three parts enhance their value. [ABSTRACT FROM AUTHOR]

The title of the chapter may suggest more than it offers. Except in two problems, there is no discussion of the partners; they are in the background, taken for granted. The emphasis is on the kinds of fields and how to divide them, given a variety of conditions. Fibonacci offered a simple introduction dividing the chapter in four parts, each named after the shape of the field to be partitioned: triangular, quadrilateral, multilateral, and circular. The Table of Contents displays the complexity as the chapter develops, the figures ranging from triangles through quadrilaterals to pentagons, circles, and parts of circles. The conditions for dividing the figures are these: in two equal parts, several equal parts, two parts in a given ratio, and several parts in a given ratio. The divisions contain additional conditions. They require drawing a line that passes through a point situated at a vertex of the figure, at a vertex or a point on a side, on a side not produced, on one of two parallel sides, on the middle of the arc of a circle, on the circumference or outside of the circle, inside the figure, outside the figure, either inside or outside the figure, either inside or outside or on a side of the figure, and in a certain part of the plane of the figure. Furthermore, the line may be drawn parallel to the base of the proposed figure or parallel to the diameter of the circle. It may also be required to draw more than one transversal, such as through one point, through two points, or lines parallel to one another in which case the problem is indeterminate. Finally, circles and their parts are also divided. [ABSTRACT FROM AUTHOR]

The longest of the eight chapters, Chapter 3 focuses on measuring fields of all geometric descriptions. Within its scope are triangles, quadrilaterals, other polygons with straight sides, circles and their parts, and oblique figures with straight and curved sides including fields along the sides of mountains. Fibonacci began the measurement of triangles in a very general way, identifying the three types of triangles and stating what is necessary to find the area of each, enough information to satisfy practitioners, but not so for theoreticians. For the latter and considering the various types of triangles, he discussed all possible ways an altitude may be drawn (he used the word cathete for altitude). A few remarks on the Pythagorean theorem, the use of Hero's formula for finding the area of a triangle given the lengths of its sides, and the method used by surveyors to measure fields rounds out the section on the measurement of triangular fields. [ABSTRACT FROM AUTHOR]

The scope of Chapter 2 is square roots of numbers: how to find them and how to compute with them. (Cube roots are discussed in Chapter 5.) Not only are the readers instructed on how to find square roots of up to six digit numbers, each clearly exemplified, but they are returned to the world of Pisan units of measurement and shown how to find the roots of square units of rod, feet, and inches, not to overlook a digression into the astronomical world of square degrees, minutes, seconds, and thirds (!). Fibonacci offered two ways of proving that a root is what it claims to be, the obvious method of multiplying a number by itself and one that suggest modular arithmetic [11]. Noteworthy is the fact that he made no reference to any of the geometric theorems discussed at length in Chapter 1, even if the reader can see where some are applied. In grappling with large numbers such as 9876543 he advised the reader to find first the root of the last five digits (98765). Next and after joining any remainder to the remaining digits (43), find its root. And then simply put the two roots together as one root. Finally, and after a practical definition of radix, the readers are instructed on how to add, subtract, multiply, and divide radicals. [ABSTRACT FROM AUTHOR]

A set of twenty-five solved problems and twelve theorems comprise the two parts or Methods of Chapter 1. In Part I all of the problems focus on finding areas of fields given dimensions in one, two, and/or three different units of measurement, which make the multiplication complex. Fibonacci's method for multiplication most probably reflects the method common to Pisa, if not much of the Mediterranean world. A crucial factor is one's ability to move rapidly among the various units, just as a modern person would be expected to move easily among the various metric or English units. [ABSTRACT FROM AUTHOR]

I. Among the various phenomena of corporeal nature, there are two, which, as they are very greatly distinguished almost before all others, having been examined in themselves, have occupied to a very great extent the philosophers of all time. One of these is that general tendency towards its centre of all bodies moving about the surface of the earth, which is commonly called gravity; the other is the regular gyration of planets in their orbits, which recurs with definite periods. Various hypotheses have been devised by various people for the explanation in mechanical terms of those phenomena. An impartial examination of these will prepare the way for explaining and developing that general law of universal gravitation, to which, it will be established, those two most noble effects are to be referred as a common foundation, even if at first sight they seem to have nothing in common; from this we will also seize the opportunity to consider along the way certain other forces of nature, which it is necessary to put in place for the solution of certain other phenomena, which philosophers have undertaken to explain likewise by mechanical theories. [ABSTRACT FROM AUTHOR]

The prize topic proposed by the Royal Academy of Sciences for 1740 was the tides: le Flux et Reflux de la Mer. The winners, in the order in which their essays were published by the Academy, were Father Antoine Cavalleri [23], Daniel Bernoulli [12], Colin MacLaurin [68] and Leonhard Euler [38]. Cavelleri's work was based on the Cartesian idea of vortices;47 the other three were founded on Newtonian principles and were subsequently reproduced in the "Jesuit Edition" of the Principia (1739-1742) [86] as illustrations of the Newtonian philosophy. Newton had already discussed in its earlier editions the tidal forces on the Earth which resulted from the gravitational attraction of the Sun and of the Moon; his work was of course the starting point for MacLaurin's investigations. [ABSTRACT FROM AUTHOR]

The main purpose of the present volume is to discuss, and to make generally accessible, three dissertations1 by the Scottish mathematician Colin MacLaurin (1698-1746), who was regarded both in Britain and in continental Europe as one of the leading mathematicians of his time. [ABSTRACT FROM AUTHOR]