Euclid's Elements of plane geometry [book 1-6] with explanatory appendix, and supplementary propositions, by W.D. Cooley1840 |
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Σελίδα 8
... Consequently , the boy of ten years and the man of forty are perfectly on a level in commencing the study of Mathematics ; for if the adult has the advantage in steady attention and the vigour of the reasoning faculty , which it is the ...
... Consequently , the boy of ten years and the man of forty are perfectly on a level in commencing the study of Mathematics ; for if the adult has the advantage in steady attention and the vigour of the reasoning faculty , which it is the ...
Σελίδα 31
... greater than CBA ; but ACD = BCG ( Prop . 15 ) ; therefore ACD also will be greater than CBA , and is consequently greater than either of the internal remote angles PROP . XVII . THEOR . Any two angles of BOOK I. PROP . XVI . 31.
... greater than CBA ; but ACD = BCG ( Prop . 15 ) ; therefore ACD also will be greater than CBA , and is consequently greater than either of the internal remote angles PROP . XVII . THEOR . Any two angles of BOOK I. PROP . XVI . 31.
Σελίδα 35
... consequently greater than EF which is equal to GC ; but BC is the base subtend- ing the greater angle in the one triangle , which is therefore greater than the base EF , which subtends the less angle in the other triangle . PROP . XXV ...
... consequently greater than EF which is equal to GC ; but BC is the base subtend- ing the greater angle in the one triangle , which is therefore greater than the base EF , which subtends the less angle in the other triangle . PROP . XXV ...
Σελίδα 46
... consequently , that GO is one straight line . And since GO and HK are parallel , the alter- nate angles HKI , KLI are equal ( Prop . 29 ) , and therefore IKM being added to both , HKI + ZIKM = / KIL + ZIKM ; but the latter are equal to ...
... consequently , that GO is one straight line . And since GO and HK are parallel , the alter- nate angles HKI , KLI are equal ( Prop . 29 ) , and therefore IKM being added to both , HKI + ZIKM = / KIL + ZIKM ; but the latter are equal to ...
Σελίδα 55
... consequently , AD2 + DB ' , or the squares of the unequal parts , are equal to 2 AC2 + 2 CD2 , or double the squares of half the line and the intermediate part . PROP . X. THEOR . If a straight line ( AB ) be bisected ( in c ) , and ...
... consequently , AD2 + DB ' , or the squares of the unequal parts , are equal to 2 AC2 + 2 CD2 , or double the squares of half the line and the intermediate part . PROP . X. THEOR . If a straight line ( AB ) be bisected ( in c ) , and ...
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.