are equiangular (Cor. 2. 34. 1.) with it. And since either of the As into which the diagonal divides the whole parallelogram is similar to one of the As into which it divides those about the diagonal, the sides about those equal as are proportional; .. &c. PROP. XXV. PROB. To construct a rectilinear figure, equal to a given one, and similar to another. To any side of the figure to which a similar one is to be constructed, apply a rectangle = to it (Cor. 1. 45. 1), and to either side of this rectangle, adjacent to the side to which it is applied, apply another rectangle = to the other given figure; and find a mean proportional between the two sides of those constructed rectangles, that are in directum; the figure constructed on this mean proportional similar to the one required, and similarly posited, will be equal to the other given one. Those two similar figures are to each other in the duplicate ratio, their homologous sides, i. e. by constr. in the ratio of the sides of the rectangles, to which a mean proportional was found, i. e. as the rectangles, i. e. as the (by to the tangles. Therefore the given figure, to which it was required to construct one similar, bears the same ratio to the other given figure and to the constructed figure. If similar and similarly posited parallelograms have a common angle, they are about the same diagonal. If possible let them not be about the same diagonal; then the diagonal of the greater parallelogram will cut a side of the less, not at the vertex of the angle which is opposite to the common angle, but in some other point; from this point, in which it cuts one of the sides of the less par., draw a right line parallel to the other side. Then the parallelogram thus formed (i. e. part of the less) will be similar to the greater (24. 6) and .. to the less, s. one side of the less bears the same ratio to two unequal right lines, which is absurd. Therefore, &c. PROPS. XXVII. XXVIII. XXIX. omitted. PROP. XXX. PROB. To cut a given finite right line in extreme and mean ratio. By extreme and mean ratio is meant, so that the whole line shall be to the greater segment as the greater segment is to the less (Def. 4. B. 6). :. the rectangle under the whole line and lesser segment shall be = to the O’ of the greater, for the greater segment is a mean proportional between the given line and lesser; if .. you cut the given line, as in Prop. XI. Book 2. the thing is done. Note 1. If this series be continued in infinitum, it is evident from Prop. 11. of this Book, that the whole line will be a mean proportional between the smaller segment and the sum of the series. And the successive terms of of the series will be found by taking each consequent from its antecedent. Cor. Prop. 11. B. 2. Note 2. It will also be seen, by the doctrine of ratios, that the successive terms of the series will be found by taking each consequent from its antecedent; for in any analogy the first term is to the second as the difference between the first and third is to the difference between the second and fourth. But when a line is cut in extreme and mean ratio, the greater segment : the lesser : : the wbole line : the greater segment; the greater segment : the lesser : : the difference between the whole and greater, i. e. as the lesser : the difference between the greater and lesser; .. the lesser segment is a mean proportional between the greater and the difference between the segments. Note 3. From those principles, we may describe on a given right line a right angled A in which the O’ of one side about the right shall be = to the rectangle under the hypothenuse and the other side, and the three sides, ... shall be in continued proportion. Note 4. It also follows that if one side of a A be divided in extreme and mean ratio, and from the point of section a right line be drawn to the opposite <, and another line parallel to the side .conterminous with the lesser segment, the two As thus cut off shall be =, and shall have the sides about their = cos reciprocally proportional. PROP. XXXI. THEOR. If on the sides of a right angled triangle any similar recti lineal figures be similarly described, the figure described on the side subtending the right angle is equal to the sum of the figures on the other two sides which contain the right angle. From the right draw a perpendicular to the hypochenuse; then the hypothenuse : one of the sides about the right <:: that side : the adjacent segment of the hypothenuse (Cor. 8. 6), the figure upon the hypothenuse : to the similar figure upon that side about the right <: : the hypoth. : the seg. ment adjacent to that side, and therefore the figure upon the hypothenuse : the difference between itself and the figure upon that side : : the hypoth. : the difference between itself and the segment adjacent to that side, i. e. ; : the hypoth, : the segment adjacent to the other side about the right %; but the figure upon the hypoth. : the similar figure upon this other side about the right 2 :: the hypoth; this difference, i.e. : : it is to the segment adjacent to this other side, .'. the figure upon this other side about the right < is = to the difference between the similar figures upon the hypothenuse and the first mentioned side about the right <. Therefore, &c. Cor Given any number of similar rectilineal figures, a similar figure can be found equal to their sum, by Cor. 2. Prop. 47. B. 1. and by Cor. 3. Prop. 47. B, 1. a figure can be found equal to the difference between two given similar figures. PROP. XXXII. THEOR. If two triangles have two sides proportional, and be so placed at an angle that the homologous sides are parallel, and that the sides not homologous form the angle at which they are placed, the remaining sides form one right line. Since the homologous sides are parallel, and the sides not homologous form an <, each of the <s contained by the proportional sides is = to this <so formed, and.. to one another;.. the given As are equiangular (Prop. 6. B. 6); therefore the three s at the point where the sides not homologous meet are together = to two right <s, they being = to the three angles of either a ; therefore, &c. Note. The given triangles will not be capable of being so placed unless they are similar. It is evident from Prop. 4. B. 6. that any two similar as may be so placed. PROP. XXXIII. THEOR. In equal circles angles, whether they be at the centres or at the circumferences, have the same ratio that the arches on which they stand have to one another; so also have the sectors. The <s at the centre are proved to be as the arcs on which the they stand in the same way that we have demonstrated the first proposition of this book; i. e. by assuming any equisubmultiples of the antecedents, i. e. of one arc and the < which stands on it, those submultiples will be contained an = number of times in their respective consequents, i. e. in the other arc and <. Equisubmultiples of the antecedents are assumed by taking any submultiple of the first arc, and drawing from its extremity a radius, this radius will cut off an equisubmultiple of the first <; for the < at the centre will be divided into as many = parts as the arc subtending it, if radii be drawn from the extremities of those parts into which the arc is divided. The ss at the centre being thus proved to be as the arcs on which they stand; it follows that the <s at the circumference are so. In the proof of that part concerning the sectors, it is necessary to prove, that the sectors of the same or equal circles standing on equal arcs are =; and .. that the radius to the extremity of the assumed submultiple of the first arc cuts off an equisubmultiple of the other antecedent ; this is thus proved ; draw chords of = arcs, these divide the sectors into triangles and segments that are respectively = to one another; the As are = by Prop. 4. or 8. B. 1; and the segments are = Prop. 24. B. 3. for if angles be inscribed in them they shall be = since they stand on = arcs, i. e. on the differences between the whole = arcs and the whole circumferences ; .. the segments are similar, and since they stand on = bases they are = (Prop. 24. 3). Therefore the sectors are = since they are made up of segments. Then whatever submultiple one of the small arcs is of the antecedent arc the same is the sector standing on this small arc of that on the antecedent; and as often as the small arc is contained in the consequent arc, so often is the small sector, &c. Cor. 1. An angle at the centre is to four right angles, as the arc upon which it stands to the whole circumference. For the < at the centre : a right <:: the arc on which it stands : {th the whole circumference (Prop. 33. 6.) and .. it is to four right <s as the arc is to the whole circumference. Cor. 2. The arcs of unequal circles, which subtend equal angles are similar. For it is evident that they have the same ratio to the whole circumferences. (Cor. 1.) Cor. 3. It is evident that the arcs of similar segments are similar. END OF THE SIXTH BOOK. |