## Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle and the Geometry of Solids ; to which are Added Elements of Plane and Spherical Trigonometry |

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**Ratio**is a mutual relation of two magnitudes , of the same kind , to one another , in respect of quantity . IV . Magnitudes are said to be of the same kind , when the less can be mul- tiplied so as to exceed the greater ; and it is only ... Page 109

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**ratio**than the third magnitude has to the fourth ; and , on the contrary , the third is said to have to the fourth a less**ratio**than the first has to the second . VIII . When there is any number of magnitudes greater than two , of ... Page 110

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**ratio**, which is compounded of two equal**ratios**, is dupli- " cate of either of these**ratios**. " XII . If four magnitudes are continual proportionals , the**ratio**of the first to the fourth is said to be triplicate of the**ratio**of the ... Page 113

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**ratio**to the second which the third has to the fourth , and if any equimultiples whatever be taken of the first and third , and any whatever of the second and fourth ; the multiple of the first shall have the same**ratio**to the ... Page 114

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**ratio**to the same magnitude ; and the same has the same**ratio**to equal magnitudes . Let A and B be equal magnitudes , and Cany other ; A : C : B : C . Let mA , mB , be any equimultiples of A and B ; and nC any multi- ple of C. Because A ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD altitude angle ABC angle ACB angle BAC angle EDF arch AC base BC bisected centre circle ABC circumference cosine cylinder definition demonstrated diameter draw equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given straight line greater hypotenuse inscribed join less Let ABC Let the straight line BC magnitudes MDCCCXX meet multiple opposite angle parallel parallelepipeds parallelogram perpendicular polygon prism PROB produced proportionals proposition Q. E. D. COR Q. E. D. PROP radius ratio rectangle contained rectilineal figure remaining angle segment semicircle shewn side BC sine solid angle solid parallelepipeds spherical angle spherical triangle straight line AC THEOR third touches the circle triangle ABC triangle DEF wherefore