Prop. 50. Prop. 44. Prop. 39. Prop.44. Join AF, draw FG_AC; .. AE. EC+EG2 = AG2, .. AE. EC+EG2+GF2 = AG2+GF2; .. AE. EC+ EF2 = AF2 or FB2, FB BE. ED+EF2, .. AE. EC+ EF2 = Take away EF2 which is common. = Lastly. Let neither AC nor BD Fig. 4. the Cr. pass through Take the Cr. F, through E the intersection of PROP. LXIII. THEOR. 36. 3 Eu. If from any point without 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 the circle, and the part of it without the circle, shall be equal to the square of the line which touches it. Let D be any point without the ABC; and DCA, DB two str. lines drawn from it, of which DCA cuts the O, and DB touches it: then AD. DC = DB2. (Figs. 1 and 2.) 1st. Let DCA pass through the Cr. E. Fig. 1. join EB; EBD rt. L Prop. 53. And AC is bisected in E and prod. to D, .. AD. DC+EC2 = ED2, Prop. 45. ... AD.DC+EB2 = EB'+DB3 take away EB3 which is common, AD. DC DB2. 2ndly. If DCA do not pass through the Cr. Take E the Cr., draw EF AC; EF which passes through the Cr. cuts AC which does not pass through the Cr. at rt. s, it also bisects it: .. AFFC; And AC is bisected in F, and prod. to D, Prop. 39. Fig. 2. Prop. 50. Prop. 39. .. AD.DC+FC2+FE2= FD2+FE2, but EFD rt. L, EC FC+FE, AD. DC+EC2 = ED2; EC = EB, .. AD. DC+EB2 = ED'; and since Prop. 39. ZEBD = rt. L, .. AD. DC+EB2 = EB2+DB3, Wherefore, if from any point, &c. COR.-If from any pt. without a , str. lines as AB, AC be drawn (fig. 3); the rectangles contained by the whole lines and the parts of them without the O, are equal to one another, viz. : BA. AE CA. AF; for each of these rectangles is equal AD3. PROP. LXIV. PROB. 5. 4 Eu. To describe a circle about a given triangle. Let the given be ABC; it is required to describe a O about the ABC, so that the may pass through the angular points A, B, C of the A. EFLAC, then DF and EF prod. will meet each other; For if not, then DF || EF, .. AB and AC, at rt. Zs to them are || which . DF and EF do meet each other. If F be not in BC, join BF, CF; AD = DB, {DF common, F, being rt. 28: base FA = base FB; similarly FC = FA, = FB FC, .. AO descr. from Cr. F, at the distance of one of them, will pass through A, B, C the angular points of the ; which was to be done. Def. 35. Prop. 4. DEF. 1.-Quantities of the same kind may be compared with each other with respect to their magnitudes, by means of the number of equal parts or units they each contain ; and ratio is the relation which two such quantities have to each other, determined by considering what fractional part one is of the other when both quantities are numerically expressed; as one line is compared with another, by means of the number of equal units, as inches, &c., and one surface with another surface, by the number of square or superficial units which they respectively contain. number of equal units contained in 2 geometrical quantities of the same kind be represented by a and b, the ratio of the quantities α If the is therefore expressed by the fraction The meaning of which is, that one quantity being divided into as many equal parts as a expresses, the other is divisible into a number of the same parts equal to b. Geometrical quantities may also be so related to each other, as to admit of no numbers that will strictly represent them, as the side and diagonal of a square, which by Prop. 39, are as 1 to 2, or 1 to 1,414, &c.; since they do not each contain an exact number of inches, or equal parts of an inch, without a remainder; yet by the subdivision of the inch (or whatever unit of measurement is used) into any number of equal parts, the ratio of the lines may be obtained to any assigned degree of accuracy. By the continued subdivision of |