If all angles be taken, beginning from one minute, and proceeding through 2', 3', &c., up to 45°, or 2700', and tables be formed by a calculation, the nature of which we cannot explain here, of their sines, cosines, and tangents, or of the logarithms of these, the proportions of every right-angled triangle, one of whose angles is an exact number of minutes, are registered. We say sines, cosines, and tangents only, because it is evident, from the table above made, that the cosecant, secant, and cotangent of any angle, are the reciprocals of its sine, cosine, and tangent, respectively. Again, the table need only include 45°, instead of the whole right angle, because, the sine of an angle above 45° being the cosine of its complement, which is less than 45°, is already registered. Now, as all rectilinear figures can be divided into triangles, and every triangle is either right-angled, or the sum or difference of two right-angled triangles, a table of this sort is ultimately a register of the proportions of all figures whatsoever. The rules for applying these tables form the subject of trigonometry, which is one of the great branches of the application of algebra to geometry. In a right-angled triangle, whose angles do not contain an exact number of minutes, the proportions may be found from the tables by the method explained in Chapter XI. of this treatise. It must be observed, that the sine, cosine, &c. are not measures of their angle; for, though the angle is given when either of them is given, yet, if the angle be increased in any proportion, the sine is not increased in the same proportion. Thus, sin 2A is not double of sin A.

The measurement of surfaces may be reduced to the measurement of rectangles; since every figure may be divided into triangles, and every triangle is half of a rectangle on the same base and altitude. The superficial unit or quantity of space, in terms of which it is chosen to express all other spaces, is perfectly arbitrary; nevertheless, a common theorem points out the convenience of choosing, as the superficial unit, the square on that line which is chosen as the linear unit. If the sides of a rectangle contain a and b units (Geometry I. 29.), the rectangle itself contains ab of the squares described on the unit. This proposition is true, even when a and b are fractional. Let the number of units in the sides be and

m n

and take anoand take ano

of the rectangles AH and AI, 'and is greater than the difference of AK and AI; therefore, the approximate fractions which represent AC and AB may be brought so near, that their product shall, as nearly as we please, represent the number of square units in their rectangle.

In precisely the same manner it may be proved, that if the unit of content or solidity be the cube described on the unit of length, the number of cu bical units in any rectangular parallelopiped, is the product of the number of linear units in its three sides, whether these numbers be whole or fractional; and in the sense just established, even if they be incommensurable with the unit.

These algebraical relations between the sides and content of a rectangle or parallelopiped were observed by the Greek geometers; but as they had no distinct science of algebra, and a very imperfect system of arithmetic, while, with them, geometry was in an advanced state; instead of applying algebra to geometry, what they knew of the first was by deduction from the last: hence the names which, to this day, are given to aa, aaa, ab, which are called the square of a, the cube of a, the rectangle of a and b. The student is thus led to imagine that he has proved that square described on the line, whose number of units is a, to contain aa square units, because he calls the latter the square of a. He must, however, recollect, that squares in algebra and geometry mean distinct things. It would be much better if he would accustom himself to call aa and aaa the second and third powers of a, by which means the confusion would be avoided. It is, nevertheless, too much to expect that a method of speaking, so commonly received, should ever be changed; all that can be done is, to point out the real connexion of the geometrical and algebraical signification. This, if once thoroughly understood, will prevent any future misconception.