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ner it may be shewn that the two triangles BOD, BOE are equal; therefore OD is equal to OC; therefore the three perpendiculars OD, OE, OF are all equal.
Now, if from the point O as a centre, with the radius OD, a circle be described, this circle will evidently be inscribed in the triangle ABC; for the side AB, being perpendicular to the radius at its extremity, is a tangent; and the same thing is true of the sides BC, AC.
Scholium. The three lines which bisect the angles of a triangle meet in the same point.
On a given straight line AB, to describe a segment containing a given angle C, that is to say, a segment such that all the angles inscribed in it shall be equal to the given angle C.
Produce AB towards D; at the point B, make the angle DBE C; draw BO perpendicular to BE, and GO perpendicular to AB, and bisecting it; and from the point Ò, where those perpendiculars meet as a centre, with a distance OB describe a circle: the required segment will be AMB.
For since BF is a perpendicular at the extremity of the radius OB, it is a tangent, and the angle ABF (19. II.) is measured by half the arc AKB. Also the angle AMB, being an inscribed angle, is measured by half the arc AKB: hence we have AMB ABF=EBD=C: hence all the angles inscribed in the segment AMB are equal to the given angle C.
Scholium. If the given angle were right, the required segment would be the semicircle described on the diameter AB.
To find the numerical ratio of two given straight lines AB, CD, those lines being supposed to have a common measure.
From the greater AB cut off a part equal to the less CD, as many times as possible; for example, twice, with the remainder BE.
From the line CD, cut off a part equal to the remainder BE, as many times as possible; once, for example, with the remainder DF.
From the first remainder BE, cut off a part equal to the second DF, as many times as possible; once, for example, with the remainder BG.
From the second remainder DF, cut off a part equal to BG the third, as many times as possible.
Continue this process, till a remainder occur, which is contained exactly a certain number of times in the preceding one.
Then this last remainder will be the common measure of the proposed lines; and regarding it as unity, we shall easily find the values of the preceding remainders; and, at last, those of the two proposed lines, and hence their ratio in numbers.
Suppose, for instance, we find GB to be contained exactly twice in FD; BG will be the common measure of the two proposed lines. Put BG-1; we shall have FD=-2: but EB contains FD once, plus GB; therefore we have EB=3: CD contains EB once, plus FD; therefore we have CD-5: and, lastly, AB contains CD twice, plus EB; therefore we have AB-13; hence the ratio of the lines is that of 13 to 5. If the line CD were taken for unity, the line AB would be 13; if AB were taken for unity, CD would be 13.
Scholium. The method just explained is the same as that employed in arithmetic to find the common divisor of two numbers it has no need, therefore, of any other demonstration.
How far soever the operation be continued, it is possible that no remainder may ever be found, which shall be contained an exact number of times in the preceding one. When this happens, the two lines have no common measure, and are said to be incommensurable. An instance of this will be seen afterwards, in the ratio of the diagonal to the side of the square. In those cases, therefore, the exact ratio in numbers cannot be found: but, by neglecting the last remainder, an approximate ratio will be obtained, more or less correct, according as the operation has been continued a greater or less number of times.
Two angles A and B being given, to find their common measure, if they have one, and by means of it their ratio in numbers. With equal radii describe the ares CD, EF, to serve as measures for the angles: proceed afterwards in the comparison of the arcs CD, EF, as in the last pro
blem, since an are may be cut off
from an arc of the same radius,
as a straight line from a straight line. We shall thus arrive at the common measure of the arcs CD, EF, if they have one, and thereby at their ratio in numbers. This ratio (17. II.) will be the same as that of the given angles; and if DO is the common measure of the arcs, DÃO will be that of the angles.
Scholium. According to this method, the absolute value of an angle may be found by comparing the arc which measures it to the whole circumference. If the arc CD, for example, is to the circumference as 3 is to 25, the angle A will be of four right angles, or of one right angle.
It may also happen that the arcs compared have no common measure; in which case, the numerical ratios of the angles will only be found approximately with more or less correctness, according as the operation has been continued a greater or less number of times.
THE PROPORTIONS OF FIGURES.
1. WE shall give the name equivalent figures to such as have equal surfaces.
Two figures may be equivalent though very dissimilar: a circle, for instance, may be equivalent to a square, a triangle to a rectangle.
The denomination, equal figures, we shall reserve for such as, when applied to each other, coincide in all their points: of this kind are two circles, which have equal radii; two triangles, which have all their sides equal respectively, &c.
II. Two figures are similar, when they have their angles equal each to each, and their homologous sides proportional. By homologous sides, are understood those which have a corresponding position in the two figures, or which lie adjacent to equal angles. Those angles themselves are called homologous angles.
Two equal figures are always similar; but two similar figures may be very unequal.
III. In two different circles, similar arcs, sectors, segments, are those which correspond to equal angles at the centre.
Thus, if the angles A and O be equal, the arc BC will be similar to the arc DE, the sector ABC to the sector ODE, &c.
IV. The altitude of a parallelogram is the perpendicular EF which measures the distance of two opposite sides AB, CD, taken as bases.
V. The altitude of a triangle is the perpendicular AD let fall from the vertex of an angle A, on the opposite side BC taken as a base.
VI. The altitude of a trapezium is the perpendicular EF drawn between its two parallel sides AB, CD.
VII. The area and the surface of a figure are terms very nearly synonimous. The area designates more particularly the superficial extent of the figure, in so far as it is measured, or compared to other surfaces.
Note. For understanding this Book, and those that follow, the reader will require to be master of the theory of proportions, for which we refer him to the common treatises on arithmetic and algebra.* We shall only make one remark, which is of great importance for determining the true meaning of propositions, and dissipating any obscurity that may occur either in the enunciations or the proofs.
The proportion A: B:: C: D being given, it is well known
A short sketch of the subject has been prefixed to the present translation.
that the product of the extremes AxD is equal to the product of the means Bx C.
This truth is indisputable, so far as concerns numbers: it is equally so in regard to magnitudes of any kind, provided they are expressed, or imagined to be expressed, in numbers; and this we are at all times entitled to imagine. If A, B, C, D, for example, are lines, we may conceive that one of those four lines, or a fifth, if requisite, serves as a common measure to them all, and is taken for unity: then A, B, C, D, represent each a certain number of units, integer or fractional, commensurable or incommensurable; and the proportion between the lines A, B, C, D, becomes a proportion of numbers.
The product of the lines A and D, which is also named their rectangle, is therefore nothing else than the number of linear units contained in A, multiplied by the number of linear units contained in D; and it is easy to see that this product may, indeed must, be equal to that which results in a similar manner from the lines B and C.
The magnitudes A and B may be of one species, for example, lines; and C and D of another species, for example, surfaces in such cases, those magnitudes must always be regarded as numbers; A and B will be expressed in linear units, C and D in superficial units; and the product AxD will be a number like the product Bx C.
Generally, in every operation connected with proportions, if the terms of those proportions be always looked upon as so many numbers, each of the kind proper to it, there will be no difficulty in conceiving the operations, and the consequences which result from them.
We must also premise, that several of our demonstrations are grounded on some of the simpler operations of algebra, which are themselves dependent on admitted axioms. Thus, if we have A-B+C, and if each member is multiplied by the same quantity M, we may infer that AxM=BxM+CXM; in like manner, if we have A=B+C, and D-E-C, and if the equal quantities are added together, then expunging the +C and C, which destroy each other, we infer that A+D=B+E, and so of others All this is evident enough of itself; but in case of difficulty, it will be useful to consult some algebraical treatise, and thus to combine the study of the