parallelograms IFHO and MFKN, of which they are also sides, and contain the supplements of the former angles, namely EFHand MF к. that is, IF:FH =MF:FK; wherefore, those two parallelograms are similar to each other; but they are also similar to the whole parallelogram OLNP; for its sides o L and op are respectively the sums of HF + FK, and E F + F M, and these sums are proportional to E F and FH, or MF and FK taken singly, proportionals being proportional by addition of their corresponding terms. Therefore, the line ON bisects the whole parallelogram o LN P, and the two parts of it, OEFHand мг км. The parallelograms which have the sum of their diameters equal to the diameter of a parallelogram containing them, and their sides parallel to its sides, are called parallelograms about the diameter of a parallelogram; and the remaining parts are called complements to these. In the case of two parallelograms about the diameter, as in the present figure, there are two complements; but whatever number there are, the corresponding complements are always equal to each other, and the parallelograms about the diameter are similar to the whole and to each other. By means of these correspondencies, it is easy to change a parallelogram to another having its sides in any proportion, by making it a complement to the given proportional; and it is equally easy to make any change in the angles without changing the area; for we have only to put the figures upon equal bases and between parallels equally distant from each other. The same facility of change applies to triangles; for a triangle is always half the parallelogram on an equal base, and between parallels equally distant from each other; and thus we can apply triangle after triangle to any given line, so as to form the whole into one parallelogram, having that line for one of its sides, and one of its angles equal to any angle that may be proposed. After we have obtained the parallelogram, we have only to change its angles or its sides by the methods already explained, in order to reduce it to a parallelogram having its sides in any proportion, or into a square. As we can perform all these operations directly we can also perform them inversely, and resolve a square into any rectangle, parallelogram, triangle, or number of triangles, that we please, or into any other straight line figure whatever; but these transformations are so seldom needed in practice, that it is unnecessary to go into any details of them; for any one who studies with attention what has been stated, can find no difficulty in performing for himself any or all of these transformations; and they involve the whole principles necessary for the comparison of every form of rectilineal figure, which is the utmost extent to which we purpose to carry the elements of Geometry, in the present volume. There is still however one determination of the relations of lines by means of the circle to which it may not be improper to advert; and that we shall now briefly consider. 3. If from any point, without the circumference of a circle, two straight lines be drawn, one of which cuts the circle and the other touches it, the rectangle contained by the whole line which cuts, and the part of that line without the circle, is equal to the square of the line which touches. The line which touches the circle is always of the same length, if the point from which it is drawn is fixed, that is, if the position of it is given; for it can touch the circle only in one point, which is fixed also because the position of the circle is given; and therefore the only variation which can be is in the position of the line which cuts the circle. It may pass through the centre of the circle, or it may not; but in what manner soever it cuts the circle, while it is drawn from the same point without the rectangle under the whole of it, and the part without the circle remains constantly the same, or equal to the square of the line which it touches; and this line itself remains unchanged, while the point in the circle remains the same. There is, therefore, the same relation between the whole of a line which cuts a circle from a point without, and the part of it which is without, that there is between the segments of a line which cuts another line in the same point within a circle, that is, the rectangle between them is of the same value however their lengths may vary, and constantly the one increases in proportion as the other diminishes; and if we take two lines drawn from the same point without, and both cutting a circle, the whole and the part without of the one must be reciprocally proportional to the whole and the part without of the other. If we show the truth of this in the case of one line which passes through the centre, and of another line which does not, we shall have proved all that is necessary. Let there be any circle of which cis the centre, and any point P without the circle. Let there be drawn from p, a line PT touching the circle in the point r, and two lines PL, PN, both cutting the circle, but P I passing through the centre and 496 LINES INTERSECTING, ETC. PN not, the rectangle under the whole p 1, and the part without the circle PD, and the rectangle under the whole PN, and the parts without ps, are each equal to the square of P T, and, consequently, they are equal to each other. First, let us consider P 1. Join T c, and because PT touches atr, and rc from the point of contact passes through the centre, prcis a right angle, and PCPT + т с2. But I D is bisected in c and produced to P, therefore But D c2 = T сo, for both lines are radii, therefore Take away т с2, which is common to both, and PIX PD = PT. Second, P XPS is also equal to PT2. Join cs, and from c draw c o at right angles to PN, and the parts N is bisected in o, therefore, PN X PS + SO2 = P02. To each of these add o co, and PN X PS + so° + oc2 = PO2 + Oc2. But PC2 = PO2 + oc2, and cso = s2 + oc2; therefore, PN X PS + Oc2 = Pc2; and Pc2 = Tc2 + TP2, and also tos c2 + PTo, take away the equals and PN XPS = PT2, that is, the rectangle under the whole line and the part without the circle, is equal to the square of the touching line, wheti the line which cuts does or does not pass through the centre. LONDON: BRADBURY AND EVANS, PRINTERS, WHITEFRIARS. |