## The Elements of Euclid with Many Additional Propositions and Explanatory NotesJ. Weale, 1860 |

### From inside the book

Results 6-10 of 89

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**wherefore**, also , the double of the third is greater than the double of the fourth ; therefore the third is greater than the fourth ; in like manner , if the first be equal to the second , or less than it , the third can be proved to ... Page 35

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**wherefore**, by the preceding case , B is to A as D is to C ; and therefore inversely , A is to B as C is to D ( d ) . SCHOLIUM . The foregoing proposition , expressed algebraically , is as follows : - THEOREM . - If A = m a , and B = m ... Page 43

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**wherefore**if G be greater than L , then G , H , K together are greater than L , M , N together ; and if equal , equal ; if less , less : but G , and G , H , K together are any equimultiples of A , and A , C , E together ; because if ... Page 47

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**wherefore**, as AG is to DK , so is AB to DE : but AG is equal to C , and DK to F ; therefore Ċ is to F , as AB is to DE . SCHOLIUM . This proposition may be algebraically expressed as follows : - THEOREM . A : B :: m . Am . B. A For = m ... Page 48

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**wherefore**C is to D , as E is to F ( c ) : again , because G , H are equimultiples of C , D , therefore C is to D , as G is to H ( a ) but it was proved that C is to D as E is to F therefore E is to F as G is to H ( c ) . But when four ...### Other editions - View all

The Elements of Euclid: With Many Additional Propositions, & Explanatory ... Euclid No preview available - 2023 |

The Elements of Euclid: With Many Additional Propositions, and Explanatory ... Euclid No preview available - 2013 |

### Common terms and phrases

AC is equal altitude angle ABC bisected circle ABCD circumference cone CONSTRUCTION contained COROLLARY cylinder DEMONSTRATION diameter divided double draw duplicate ratio EFGH equal angles equal in area equiangular equilateral equimultiples Euclid external angle fore fourth given line given rectilineal given straight line gnomon greater ratio homologous sides Hypoth HYPOTHESES inscribed join less line AC lines be drawn meet multiple opposite angle parallel parallelogram perpendicular polygon prism proposition pyramid ABCG pyramid DEFH rectangle rectilineal figure remaining angle right angles SCHOLIA SCHOLIUM segment side AC solid angle solid CD solid parallelopipeds sphere square on AB square on AC syllogism THEOREM THEOREM.-If third three plane angles tiple triangle ABC triplicate ratio vertex wherefore

### Popular passages

Page 107 - A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.

Page 85 - ... have an angle of the one equal to an angle of the other, and the sides about those angles reciprocally proportional, are equal to une another.

Page 18 - Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding...

Page 82 - From the point A draw a straight line AC, making any angle with AB ; and in AC take any point D, and take AC the same multiple of AD, that AB is of the part which is to be cut off from it : join BC, and draw DE parallel to it : then AE is the part required to be cut off. Because ED is parallel to one of the sides of the triangle ABC, viz. to BC ; as CD is to DA, so is (2.

Page 111 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.

Page 116 - ... plane, from a given point above it. Let A be the given point above the plane BH; it is required to draw from the point A a straight line perpendicular to the plane BH.

Page 115 - For the same reason, CD is likewise at right angles to the plane HGK. Therefore AB, CD are each of them at right angles to the plane HGK.

Page 49 - IF magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately ; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD...

Page 32 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 34 - Take of B and D any equimultiples whatever E and F; and of A and C any equimultiples whatever G and H. First, let E be greater than G, then G is less than E: and because A is to B, as C is to D, (hyp.) and of A and C...