## Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |

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Página 94

... ( iv . Prop . 1 ) , an equilateral and equiangular quindecagon will be thus inscribed in the circle . D Note . For Remarks on the Third and

... ( iv . Prop . 1 ) , an equilateral and equiangular quindecagon will be thus inscribed in the circle . D Note . For Remarks on the Third and

**Fourth**Books , see APPENDIX D. [ 95 ] BOOK V. DEFINITIONS . 1. A less 94 EUCLID'S ELEMENTS . Página 95

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**fourth**, when , any equimultiples whatsoever being taken of the first and third , and any equimultiples whatsoever of the second and**fourth**, the multiple of the third is greater than , equal to , or less than the multiple of the**fourth**... Página 96

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**fourth**, and , on the contrary , the third is said to have to the**fourth**a less ratio than the first has to the second . 8. When , in a series of magnitudes , each magnitude has the same ratio to that which immediately follows it ... Página 97

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**fourth**. ] 13. In proportionals , the antecedent terms are said to be homologous to one another , as are also the con- sequents . [ Thus , in the analogy A : B :: C : D , A and C are homo- logous terms , as are also B and D. ] 14. The ... Página 100

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**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 multiple of the second , that the multiple of the third ...### Términos y frases comunes

ACDB adjacent angles angles equal antecedent Axioms base bisected centre chord circumference coincide consequently Const definition demonstrated describe diagonal diameter difference divided draw equal angles equal Prop equal sides equiangular equilateral triangle equimultiples Euclid Euclid's Elements external angle extremities fore fourth fractional Geometry given angle given circle given line given point given straight line given triangle greater hypotenuse inscribed internal intersect isosceles triangle less line drawn lines be drawn magnitudes manner meeting multiple opposite angles parallel parallelogram perpendicular point of contact PROB produced proportional Proposition quadrilateral figure rectangle contained rectilinear figure remaining angles respectively equal right angle segment semiperimeter sides AC sides equal square of half subtending taken tangent THEOR third triangles ABC unequal vertex whole line

### Pasajes populares

Página 126 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.

Página 155 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.

Página 83 - If a straight line touch a circle, and from the point of contact a chord be drawn, the angles which this chord makes with the tangent are equal to the angles in the alternate segments.

Página 129 - ... figures are to one another in the duplicate ratio of their homologous sides.

Página 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.

Página 90 - BFE : (i. def. 10.) therefore, in the two triangles, EAF, EBF, there are two angles in the one equal to two angles in the other, each to each ; and the side EF, which is opposite to one of the equal angles in each, is common to both ; therefore the other sides are equal ; (i.

Página 117 - A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.

Página 56 - If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced.

Página 60 - Iff 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.

Página 78 - Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another. If it be possible. let the two similar segments of circles, viz. ACB' ADB be upon the same side of the same straight line AB, not coinciding with one another.