## Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's and Playfair's Editions ... |

### From inside the book

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**equimultiples**of the first and third , also of the second and fourth , being taken ; if the multi- ple of the first be greater than that of the second , the multiple of the third is greater than that of the fourth ; if equal , equal ... Page 70

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**Equimultiples**of the same , or of equal magnitudes , are equal to one another . 2. Those magnitudes are equal to one another , of which the same , or equal magnitudes , are**equimultiples**. 3. A multiple of a greater magnitude is greater ... Page 71

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**equi- multiples**whatever . Also , if there were three , four , or more magnitudes ,**equimultiples**of as many parts ; the sum of all the magnitudes would be the same multiple of the sum of all the parts that each magnitude would be of ... Page 72

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**equimultiples**be taken , these shall be**equimultiples**, one of the second the other of the fourth . Let A , B , C , D be four magnitudes , in order ; F such that A is the second multiple of B , and C the second multiple of D ; and if EF ... Page 73

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**equimultiples**of CD , and are equal to each other ( a ) . Take AE from both ; then AG and EB are left equal ( b ) ...**equimultiples**of other two ; and if**equimultiples**of the latter be taken from the former ; the parts left , if any ...### Other editions - View all

Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |

Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |

### Common terms and phrases

ABCD alternate angles angle ACD angles ABC angles equal antecedents Argument base BC bisected centre Chart chord circle ABC circumference Constr Denison Olmsted diameter draw drawn equal angles equal arcs equal radii equal sides equals the squares equiangular equilateral equilateral polygon equimultiples exterior angle fore Geometry given circle given rectilineal given straight line given triangle gles gnomon greater inscribed isosceles isosceles triangle join less meet multiple opposite angles parallelogram parallelopipeds pentagon perimeter perpendicular plane polygon produced propositions Q. E. D. Recite radius ratio rectangle rectangle contained rectilineal figure School secant segment semicircle similar sine square of AC tangent third touches the circle triangle ABC unequal Wherefore

### Popular passages

Page 90 - If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

Page 117 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.

Page 92 - 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 79 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.

Page 87 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally...

Page 26 - Triangles upon equal bases, and between the same parallels, are equal to one another.

Page 94 - Equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional ; and parallelograms that have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

Page 12 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Page 133 - If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.

Page 13 - AB be the greater, and from it cut (3. 1.) off DB equal to AC the less, and join DC ; therefore, because A in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides DB, BC are equal to the two AC, CB. each to each ; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is< equal to the triangle (4. 1.) ACB, the less to 'the greater; which is absurd.