 | W. PEASE - 1846 - 86 ページ
...another, AC will be the homologous side of a similar polygon, equal to the sum of the given polygons. For "universally, similar rectilineal figures are to one...in the duplicate ratio of their homologous sides." Duplicate ratio is the ratio of the square of one quantity to the square of another. EXAMPLES. 1. Make... | |
 | Dennis M'Curdy - 1846 - 166 ページ
...(b) p. 32, 1 ; (c) p. 4, 6 ; ( d) p. 22, 5 ; (c) def. 1, 6 and def. 35, 1. 19 Th. Similar triangles are to one another in the duplicate ratio of their homologous sides. Given the similar triangles ABC, DEF; having the angles at B, E, equal, and AB to BC as DE to EF: then... | |
 | Euclides - 1846 - 292 ページ
...And, in like manner, it may be proved, that similar figures of any number of sides more than three are to one another in the duplicate ratio of their homologous sides ; and it has already been proved (9. 19) in the case of triangles. Wherefore, universally, Similar... | |
 | Euclides - 1846 - 272 ページ
...AEDCB) may be divided into similar triangles, equal in number, and homologous to all. And the polygons are to one another in the duplicate ratio of their homologous sides. PART 1. — Because in the triangles FGI and AED, the angles G and E are G ( equal, and the sides about... | |
 | Joseph Denison - 1846 - 106 ページ
...similar, and consequently the approximating sides homologous, and (6 Euclid 19) because similar triangles are to one another in the duplicate ratio of their homologous sides; the evanescent triangles are in the duplicate ratio of the homologous sides; and this seems the proper... | |
 | Samuel Hunter Christie - 1847 - 172 ページ
...of the ratios of their bases and altitudes : the bases being similar rectilineal figures (Def. 13) are to one another in the duplicate ratio of their homologous sides (VI. 20); and the solids being similar, their altitudes are in the simple ratio of the homologous sides:... | |
 | Anthony Nesbit - 1847 - 492 ページ
...Quantity of Land, by a Line parallel to any one of its Sides. RULE. — The areas of similar triangles are to one another in the duplicate ratio of their homologous sides : hence, as the area of the triangle ABC is to the square of the side AC, or BC, so is the area of... | |
 | Thomas Gaskin - 1847 - 301 ページ
...you recollect. How did Legendre escape the difficulty by an analytical process. 2. Similar triangles are to one another in the duplicate ratio of their homologous sides. 3. If a straight line be at right angles to a plane, every plane which passes through it is at right... | |
 | Euclides - 1848 - 52 ページ
...similar, and similarly situated, to a given rectilineal figure. PROP. XIX. THEOREM. Similar triangles are to one another in the duplicate ratio of their homologous sides. COR. From this it is manifest, that if three straight lines be proportionals, as the first is to the third,... | |
 | J. Goodall, W. Hammond - 1848 - 390 ページ
...angle. (The first case only of this proposition need be demonstrated.) Section 2. 1. Similar triangles are to one another in the duplicate ratio of their homologous sides. 2. If one angle of a triangle be equal to the sum of the other two, the greatest side is double of... | |
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Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides. 
