Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
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Page 24
Euclides William Desborough Cooley. PROP . III . PROB . From the greater ( AB ) of two given straight lines , to cut off a part equal to the less ( c ) . From either extremity of AB , as A , draw a straight line AD , equal to C ( Prop ...
Euclides William Desborough Cooley. PROP . III . PROB . From the greater ( AB ) of two given straight lines , to cut off a part equal to the less ( c ) . From either extremity of AB , as A , draw a straight line AD , equal to C ( Prop ...
Page 25
... PROP . V. THEOR . In an isosceles triangle ( ABC ) the internal angles at the base are equal , and when the equal sides ( AB , AC ) are produced , the external angles at the base are also equal . Produce the equal sides AB , AC ...
... PROP . V. THEOR . In an isosceles triangle ( ABC ) the internal angles at the base are equal , and when the equal sides ( AB , AC ) are produced , the external angles at the base are also equal . Produce the equal sides AB , AC ...
Page 26
... ( Prop . 4 ) . But the triangle DCB is part of ABC , and is therefore also less than it ( Ax . 9 ) , which is absurd . There- fore DC cannot be equal to AB : and the same may be de- monstrated of any other part of AC , which is therefore ...
... ( Prop . 4 ) . But the triangle DCB is part of ABC , and is therefore also less than it ( Ax . 9 ) , which is absurd . There- fore DC cannot be equal to AB : and the same may be de- monstrated of any other part of AC , which is therefore ...
Page 27
... ( Prop . 5 ) ; but ECD is greater than BCD , which is but a part of it ( Ax . 9 ) , A and therefore FDC also must be greater than Again , since BD = BC ( Hyp . ) , ≤ BDC = / BCD ( Prop . 5. ) : but it has been proved that FDC is greater ...
... ( Prop . 5 ) ; but ECD is greater than BCD , which is but a part of it ( Ax . 9 ) , A and therefore FDC also must be greater than Again , since BD = BC ( Hyp . ) , ≤ BDC = / BCD ( Prop . 5. ) : but it has been proved that FDC is greater ...
Page 28
... ( Prop . 4 ) . ] PROP . IX . PROB . To bisect a given rectilinear angle ( BAC ) . A In the leg AB of the given angle take any point D ; cut off from the other leg AC the part AE equal to AD ( Prop . 3 ) ; draw the straight line DE , and ...
... ( Prop . 4 ) . ] PROP . IX . PROB . To bisect a given rectilinear angle ( BAC ) . A In the leg AB of the given angle take any point D ; cut off from the other leg AC the part AE equal to AD ( Prop . 3 ) ; draw the straight line DE , and ...
Common terms and phrases
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
Popular passages
Page 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'.
Page 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.
Page 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.
Page 129 - ... figures are to one another in the duplicate ratio of their homologous sides.
Page 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 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.
Page 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.
Page 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.
Page 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.
Page 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.