## 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 sider |

### Fra bogen

Resultater 1-5 af 79

Side 132

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**ratio**to the same than the less has ; and the same magnitude has a greater**ratio**to the less , than it has to the greater . Fig . 1 . E Let AB ,**BC**be unequal magnitudes , of which AB is the greater , and let D be any magnitude whatever ... Side 133

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**ratio**than**BC**has to D. Also D has to**BC**a greater**ratio**than it has to AB : for , having made the same construction , it may be shown , in like manner , that L is great- er than FG , but that it C A โซ่ L K H D K Db 7. def . 5 . is not ... Side 157

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**ratio**which their equimul- tiples haved ; as the triangle ABC is to the triangle ACD , soc 41. 1 . is the parallelogram EC to the parallelogram CF : and because d 15. 5 . it has been shown , that , as the base**BC**is to the base CD , so ... Side 158

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**ratio**to the triangle ADE ; and therefore f the triangle BDE is equal to the triangle CDE : and they are on the same base DE ; but equal triangles on the same base are between the same para- llels ; therefore DE is parallel to**BC**... Side 162

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**BC**, so is GE to EF ; but as AB to**BC**, so is DE to EF ; there- fore as DE to EF , sod GE to EF : therefore DE and GE have the same**ratio**to EF , and consequently are equale : for the same reason , DF is equal to FG : and because in the ...### Almindelige termer og sætninger

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

### Populære passager

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.