It is necessary to the understanding of this work that the reader should have a knowledge of the theory of proportions which is explained in common treatises either of arithmetic or algebra; he is supposed also to be acquainted with the first rules of algebra; such as the addition and subtraction of quantities, and the most simple operations belonging to equations of the first degree. The ancients, who had not a knowledge of algebra, supplied the want of it by reasoning and by the use of proportions which they managed with great dexterity. As for us, who have this instrument in addition to what they possessed, we should do wrong not to make use of it, if any new facilities are to be derived from it. I have accordingly not hesitated to employ the signs and operations of algebra, when I have thought it necessary, but I have guarded against involving in difficult operations what ought by its nature to be simple; and all the use I have made of algebra in these elements, consists as I have already said, in a few very simple rules, which may be understood almost without suspecting that they belong to algebra. Besides, it has appeared to me, that, if the study of geometry ought to be preceded by certain lessons in algebra, it would be not less advantageous to carry on the study of these two sciences together, and to intermix them as much as possible. According as we advance in geometry, we find it necessary to combine. together a greater number of relations, and algebra may be of great service in conducting us to our conclusions by the readiest and most easy method. This work is divided into eight sections, four of which treat of plane geometry, and four of solid geometry. The first section, entitled first principles, &c. contains the properties of straight lines which meet those of perpendiculars, the theorem upon the sum of the angles of a triangle, the theory of parallel lines, &c. The second section, entitled the circle, treats of the most simple properties of the circle, and those of chords, of tangents, and of the measure of angles by the arcs of a circle. These two sections are followed by the resolution of certain problems relating to the construction of figures. The third section, entitled the proportions of figures, contains the measure of surfaces, their comparison, the properties of a right-angled triangle, those of equiangular triangles, of similar figures, &c. We shall be found fault with perhaps for having blended the properties of lines with those of surfaces; but in this we have followed pretty nearly the example of Euclid, and this order cannot fail of being good, if the propositions are well connected together. This section also is followed by a series of problems relating to the objects of which it treats. The fourth section treats of regular polygons and of the measure of the circle. Two lemmas are employed as the basis of this measure, which is otherwise demonstrated after the manner of Archimedes. We have then given two methods of approximation for squaring the circle, one of which is that of James Gregory. This section is followed by an appendix, in which we have demonstrated that the circle is greater than any rectilineal figure of the same perimeter. The first section of the second part contains the properties of planes and of solid angles. This part is very necessary for the understanding of solids and of figures in which different planes are considered. We have endeavoured to render it more clear and more rigorous than it is in common works. The second section of the second part treats of polyedrons and of their measure. This section will be found to be very different from that relating to the same subject in other treatises; we have thought we ought to present it in a manner entirely new. The third section of this part is an abridged treatise on the sphere and spherical triangles. This treatise does not ordinarily make a part of the elements of geometry; still we have thought it proper to consider so much of it as may form an introduction to spherical trigonometry. The fourth section of the second part treats of the three round bodies, which are the sphere, the cone, and the cylinder. The measure of the surfaces and solidities of these bodies is determined by a method analogous to that of Archimedes, and founded, as to surfaces, upon the same principles, which we have endeavoured to demonstrate under the name of preliminary lemmas. At the end of this section is added an appendix to the third section of the second part on spherical isoperimetrical polygons; and an appendix to the second and third sections of this part on the regular polyedrons. INTRODUCTION. In order to abridge the language of geometry particular signs are substituted for the words which most frequently occur; and when we are employed upon any number or magnitude without considering its particular value, but merely with a view to indicate its relation to other magnitudes, or the operations to which it is to be subjected, we distinguish it by a letter of the alphabet, which thus becomes an abridged name for this magnitude. I. signifies plus, or added to. The expression A+B indicates the sum which results from the magnitude represented by the letter A being added to that represented by B, or A plus B. signifies minus. A-B denotes what remains after the magnitude represented by B has been subtracted from that represented by A. × signifies multiplied by. Ax B indicates the product arising from the magnitude represented by A being multiplied by the magnitude represented by B, or A multiplied by B. This product is also sometimes denoted by writing the letters one after the other without any sign, thus AB signifies the same as A × B. × The expression Ax (B+ CD) represents the product of A by the quantity B+C-D, the magnitudes included within the parenthesis being considered as one quantity. A B indicates the quotient arising from the magnitude represented by A being divided by that represented by B, or A divided by B. A=B signifies that the magnitude represented by A is equal to that represented by B, or A equal to B. A> B signifies that the magnitude represented by A exceeds that represented by B, or A greater than B. 2A, 3A, &c., indicate double, triple, &c., of the magnitude represented by A. II. When a number is multiplied by itself, the result is the second power, or square, of this number; 5 × 5, or 25, is the second power, or square, of 5. The second power therefore is the product of two equal factors; each of these factors is the square root of the product; 5 is the square root of 25. If the second power be multiplied by its root, the result is the third power, or cube; 5 × 25, or 125, is the third power of 5. The third power is a product formed by the multiplication of three equal factors; each of these factors is the cube root of this product; 125 is the product of 5 multiplied twice by itself, or 5 × 5 × 5; and 5 is the cube root of 125. In general A2, being an abbreviation of Ax A, indicates the second power or square of A. √ indicates the square root of A, or the number, which being multiplied by itself, produces the number represented by A. A3, being an abbreviation of AX AX A, indicates the third power or cube of A. 3 ✔ indicates the cube root of A, or the number which, being multiplied twice by itself, produces the number A. The square of a line AB is denoted by AB. The square root of a product A × B is represented by A× B. AH numbers are not perfect squares or perfect cubes, that is, they have not square roots or cube roots which can be exactly expressed; 19, for example, as it is between 16, the square of 4, and 25, the square of 5, has for its root a number comprehended between 4 and 5, but which cannot be exactly assigned. In like manner 89, which is between 64, the cube of 4, and 125, the cube of 5, has for its cube root a number between 4 and 5, but which cannot be exactly assigned. Algebra furnishes methods for approximating, as nearly as we please, the roots of numbers which are not perfect powers. III. 1. When two proportions have a common ratio, it is evident that the two other ratios may be put into a proportion, since they are each equal to that which is common. If, for example, we have |