## Euclid's Elements [book 1-6] with corrections, by J.R. Young1838 |

### From inside the book

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Page 136

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**multiple**of the less ; and the less is in this case called a submultiple of the greater , or a measure of the greater . III . Magnitudes which have a common measure , that is , which are**multiples**of any other quantity , are said to be ... Page 137

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**multiples**the same number of times . VI . When two magnitudes are compared together , in reference to the enquiry whether the first is contained in the second , or in a**multiple**of the second , the former is called an antecedent , and ... Page 138

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**multiple**of its consequent oftener than the other antecedent is contained in a like**multiple**of its consequent - is comprehensive of the common notion of proportion ; which notion , however , does not take in incommen- surable ... Page 140

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**multiple**of a greater magnitude exceeds a like**multiple**of a less ; and a submultiple of a greater exceeds a like submultiple of a less . III . That magnitude of which a**multiple**is greater than a like**multiple**of another , is greater ... Page 141

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**multiple**soever any one of the first is of its part , the same**multiple**is the sum of all the former to the sum of all the latter . First let there be two magnitudes P , Q , equimultiples of two others A , B ; then will P + Q be a like ...### Other editions - View all

### Common terms and phrases

ABCD adjacent angles alternate angles angle ABC angle ACB angle BAC angle BCD angle EDF angles equal antecedent arc BC base BC BC is equal bisected centre circle ABC circumference consequent Const demonstrated described diameter double draw equal angles equal to AC equiangular equilateral and equiangular equimultiples Euclid exterior angle fore Geometry given circle given straight line gnomon greater inscribed join less Let ABC Let the straight logarithm multiple opposite angle parallel parallelogram pentagon perpendicular PROB proportion proposition Q. E. D. PROP radius rectangle contained rectilineal figure remaining angle segment side BC similar sine square of AC straight line AB straight line AC tangent THEOR touches the circle triangle ABC triangle DEF twice the rectangle wherefore

### Popular passages

Page 30 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.

Page 105 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Page 50 - 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 61 - If a straight line be divided into two equal parts, and also into two unequal parts ; the rectangle contained by the unequal parts, together with the square on the line between the points of section, is equal to the square on half the line.

Page 65 - If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 70 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Page 41 - 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 172 - 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 45 - TRIANGLES upon the same base, and between the same parallels, are equal to one another.

Page 38 - If a, straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.