## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth. The Errors by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected, and Some of Euclid's Demonstrations are Restored. Also, the Book of Euclid's Data, in Like Manner CorrectedConrad and Company, 1810 - 518 páginas |

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Página 367

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**ratio**is said to be**given**, when a**ratio**of a**given**magnitude to a**given**magnitude which is the same**ratio**with it can be found . III . Rectilineal figures are said to be**given**in species , which have each of their angles**given**, and ... Página 368

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**given**magni- tude , when this**given**magnitude being added to it , the whole is equal to the other magnitude . PROPOSITION I. THE**ratios**of**given**magnitudes to one another is**given**. Let A , B be two**given**magnitudes , the**ratio**of A to B is ... Página 370

... given , and also the second together with the third ; either the first is equal to the third , or one of them is ... ratio to a part of it , it shall also have a

... given , and also the second together with the third ; either the first is equal to the third , or one of them is ... ratio to a part of it , it shall also have a

**given ratio**to the remaining part of it . Let the magnitude AB have a given ... Página 371

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**given ratio**to one another : because as AB to BC , so is DE to EF ; by division 4 , AC is to CB , as DF to FE : and DF , FE are given ; d 17. 5 . therefore a the ratio of AC to CB is given . PROP . VII . a 2. def . 6 . IF two magnitudes ... Página 372

... ratio of the given magnitudes DE , EF : and because the given magnitude AB has to BC the

... ratio of the given magnitudes DE , EF : and because the given magnitude AB has to BC the

**given ratio**of DE to EF , if unto DE , EF , AB a fourth proportion- D al can be found , this which is BC is given ; and because AB is given , the ...### Otras ediciones - Ver todas

### Términos y frases comunes

altitude angle ABC angle BAC base BC BC is equal BC is given bisected Book XI centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gnomon greater join less Let ABC multiple opposite parallel parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of BC rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore

### Pasajes populares

Página 17 - FG; then, upon the same base EF, and upon the same side of it, there can be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise their sides terminated in the other extremity: But this is impossible (i.

Página 35 - 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.

Página 67 - Ir any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle. Let ABC be a circle, and A, B any two points in the circumference ; the straight line drawn from A to B shall fall within the circle.

Página 92 - IF a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle, shall be equal to the angles which are in the alternate segments of the circle.

Página 26 - If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

Página 55 - 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 of the whole and that part.

Página 318 - 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.

Página 22 - If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

Página 161 - 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.

Página 21 - When a straight line standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.