Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
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Σελίδα 163
... fractional , or involve a fraction , then its square , containing the square of the fractional part , must be also fractional . The square of 41 , for example , is composed of the square of 4 = 16 , twice the product 4x = 4 , and the ...
... fractional , or involve a fraction , then its square , containing the square of the fractional part , must be also fractional . The square of 41 , for example , is composed of the square of 4 = 16 , twice the product 4x = 4 , and the ...
Σελίδα 164
... fractional , and the fraction . involved in the double product ( or 10 ) is to be added to the fractional square ( ) . Hence it might be supposed that these added fractions may sometimes amount to unity , or that the fraction may ...
... fractional , and the fraction . involved in the double product ( or 10 ) is to be added to the fractional square ( ) . Hence it might be supposed that these added fractions may sometimes amount to unity , or that the fraction may ...
Σελίδα 168
... fractional root must be also fractional . To understand more clearly the point here sought to be demonstrated , let us again have recourse to numerical illustration . The squares of the frac- tional roots 33 and 4 are each composed of ...
... fractional root must be also fractional . To understand more clearly the point here sought to be demonstrated , let us again have recourse to numerical illustration . The squares of the frac- tional roots 33 and 4 are each composed of ...
Σελίδα 169
... fractional numbers must be always fractional . Propositions 12 and 13 may be easily de- rived together from a common principle , namely , that if from any angle of a triangle a perpen- dicular be drawn to the opposite side , the dif ...
... fractional numbers must be always fractional . Propositions 12 and 13 may be easily de- rived together from a common principle , namely , that if from any angle of a triangle a perpen- dicular be drawn to the opposite side , the dif ...
Σελίδα 177
... fractional number ; while this number expressing how often one quantity measures another , is the same , the ratio is the same : thus , 1 : 3 , 2 : 6 , 3 : 9 , are all equal ratios , for all the antecedents are contained in their ...
... fractional number ; while this number expressing how often one quantity measures another , is the same , the ratio is the same : thus , 1 : 3 , 2 : 6 , 3 : 9 , are all equal ratios , for all the antecedents are contained in their ...
Συχνά εμφανιζόμενοι όροι και φράσεις
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
Δημοφιλή αποσπάσματα
Σελίδα 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'.
Σελίδα 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.
Σελίδα 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.
Σελίδα 129 - ... figures are to one another in the duplicate ratio of their homologous sides.
Σελίδα 47 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.
Σελίδα 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.