## The Elements of Euclid; viz. the first six books, together with the eleventh and twelfth. Also the book of Euclid's Data. By R. Simson. To which is added, A treatise on the construction of the trigonometrical canon [by J. Christison] and A concise account of logarithms [by A. Robertson]. |

### From inside the book

Page i

THE TWENTY -

THE TWENTY -

**FOURTH**EDITION , CAREFULLY REVISED AND CORRECTED . LONDON : LONGMAN , REES , ORME , BROWN , GREEN , AND LONGMAN ; T. CADELL ; J. , RICHARDSON ; J. M. RICHARDSON ; J. G. AND F. RIVINGTON ; BALDWIN AND CRADOCK ; E. WILLIAMS ... Page 100

Magnitudes are said to have a ratio to one another , when the less can be multiplied so as to exceed the other . V. The first of four magnitudes is said to have the same ratio to the second , which the third has to the

Magnitudes are said to have a ratio to one another , when the less can be multiplied so as to exceed the other . V. The first of four magnitudes is said to have the same ratio to the second , which the third has to the

**fourth**, when any ... Page 101

of the

of the

**fourth**: or , if the multiple of the first be equal to that of the second , the multiple of the third is also ... are proportionals , it is usually expressed by saying , the first is to the second , as the third to the**fourth**. Page 102

This word is used when there are four proportionals , and it is inferred that the first has the same ratio to the third , which the second has to the

This word is used when there are four proportionals , and it is inferred that the first has the same ratio to the third , which the second has to the

**fourth**; or , that the first is to the third , as the second to the**fourth**; as is ... Page 103

tionals , and it is inferred , that the excess of the first above the second , is to the second , as the excess of the third above the

tionals , and it is inferred , that the excess of the first above the second , is to the second , as the excess of the third above the

**fourth**, is to the**fourth**. 17th Prop . Book 5 . XVII . Convertendo , by conversion ; when there are ...### What people are saying - Write a review

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### Common terms and phrases

added altitude angle ABC angle BAC base Book centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter difference divided double draw drawn equal equal angles equiangular equimultiples Euclid excess fore four fourth given angle given in position given in species given magnitude given ratio given straight line greater Greek half join less likewise logarithm manner meet multiple opposite parallel parallelogram pass perpendicular plane prism produced Prop proportionals PROPOSITION proved pyramid Q. E. D. PROPOSITION radius reason rectangle rectangle contained remaining right angles segment shewn sides similar sine solid solid angle sphere square square of BC taken third triangle ABC wherefore whole

### Popular passages

Page 32 - To a given straight line, to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle...

Page 138 - 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 39 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 22 - If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another...

Page 41 - If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together •with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

Page 5 - If two triangles have two sides of the one equal to two sides of the other, each to each, but the...

Page 38 - IF a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Let the straight line AB be divided...

Page 262 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Page 89 - PBOR. —To describe an isosceles triangle, having each of the angles at the base, double of the third angle. Take any straight...

Page 165 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical 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 on the straight line which bisects the angle.