## The figures of Euclid with the enunciations, as printed in Euclid's Elements of plane geometry [book 1-4, 6] by W.D. Cooley1840 |

### From inside the book

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Page 56

Euclides William Desborough Cooley. PROP . IV . THEOR . The sides about the angles of equiangular tri- angles are proportional ; and those which are opposite to the equal angles are

Euclides William Desborough Cooley. PROP . IV . THEOR . The sides about the angles of equiangular tri- angles are proportional ; and those which are opposite to the equal angles are

**homologous sides**, that is , are the antecedents or ... Page 57

... sides about the equal angles proportional , the tri- angles shall be equiangular , and shall have those angles equal which are opposite to the

... sides about the equal angles proportional , the tri- angles shall be equiangular , and shall have those angles equal which are opposite to the

**homologous sides**, E G B PROP . VII . THEOR . If two triangles have one angle of the one ... Page 62

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**homologous sides**. E F B C COR . - Hence it is manifest that if three straight lines be proportional , as the first is to the third , so is any triangle upon the first , to a similar and similarly described triangle upon the second ... Page 63

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**homologous sides**. COR . 2. - If , to AB , FG**homologous sides**of two similar figures , a third proportional M be taken , it is plain that the figure on AB is to a similar and similarly placed figure on FG , as AB is to M. PROP . XXI ... Page 69

... sides containing the right angle . B PROP . XXXII . THEOR . If two triangles , having two sides of the one proportional to two sides of the other , be joined at one angle , so that their

... sides containing the right angle . B PROP . XXXII . THEOR . If two triangles , having two sides of the one proportional to two sides of the other , be joined at one angle , so that their

**homologous sides**be parallel to each other ...### Other editions - View all

The Figures of Euclid with the Enunciations, as Printed in Euclid's Elements ... Euclides No preview available - 2016 |

### Common terms and phrases

angle contained angle subtended bisect a given centre or circumference CHANCERY LANE COR.-Hence cutting line describe a circle divide a given E PROP equal angles reciprocally equal arches equal circles equal sides equal straight lines equal to four equi equilateral and equiangular EUCLID'S ELEMENTS external segment F B PROP F PROP finite straight line four straight lines given circle given finite straight given point given rectilinear angle given rectilinear figure given straight line given triangle half the line homologous sides inscribe an equilateral internal angles isosceles triangle line be bisected line be divided lines be drawn lines be proportional obtuse angle opposite angle parallelogram equal perpendicular point of contact points of section polygons PROB rectangle contained right angles side opposite side subtending Similar triangles spectively equal square of half straight line drawn straight line touches tangent THEOR three given straight touches a circle twice the rectangle whole line XXXIII

### Popular passages

Page 62 - Similar triangles are to one another in the duplicate ratio of their homologous sides.

Page 29 - In every triangle, the square of the side subtending any of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

Page 57 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.

Page 55 - IF an angle of a triangle be bisected by a straight line, which likewise cuts the base ; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the...

Page 58 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

Page 26 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C : the squares of AB, BC shall be equal to twice the rectangle AB, BC, together with the square of AC.

Page 54 - IF a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those produced, proportionally ; and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle...

Page 56 - The sides about the equal angles of equiangular triangles are proportionals ; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios Let ABC, DCE, be equiangular triangles, having the angle ABC equal to the angle DCE, and the angle ACB to the angle DEC, and consequently (4.

Page 40 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.

Page 69 - IN right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar and similarly described figures upon the sides containing the right angle...