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 have thought we ought to present it in a manner entirely new.

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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.

A× 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 A x A, indicates the second power or square of A.

V 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 A× A× A, indicates the third power or cube of A.

3

A indicates the cube root of A, or the number which, being multiplied twice by itself, produces the number A.

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The square of a line AB is denoted by AB.

The square root of a product A × B is represented by √Ã× B. All 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

2. When two proportions have the same antecedents, the con. sequents may be put into a proportion; for, if we have

A: B : : C: D,

A: E:: C: F,

by changing the place of the means, these proportions will be

IV. Other changes, besides the transposition of terms, may be made among proportionals without destroying the equality of the product of the extremes to that of the means.

1. If to the consequent of a ratio we add the antecedent, and compare this sum with the antecedent, this last will be contained once more than it was in the first consequent; the new ratio then will be equal to the primitive ratio increased by unity. If the same operation be performed upon the two ratios of a proportion, there will evidently result from it two new ratios equal to each other, and consequently a new proportion.

Let there be, for example, the proportion

2. If from the consequent of a ratio we subtract the antecedent, and compare the difference with the antecedent, this last will be contained once less than it was in the first consequent ; the new ratio will be equal to the primitive ratio diminished by unity. If the same operation be performed upon the two ratios of a proportion, there will result from it two new ratios equal to each other, and consequently a new proportion.