Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's and Playfair's Editions ... |
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Σελίδα 69
Ratio is the numerical relation of antecedent and consequent , or the number of
times , or parts of times , which the latter contains the former . Or , Ratio is the
numerical relation of measure and magnitude , or the number of times which the
...
Ratio is the numerical relation of antecedent and consequent , or the number of
times , or parts of times , which the latter contains the former . Or , Ratio is the
numerical relation of measure and magnitude , or the number of times which the
...
Σελίδα 76
8 Th . Of unequal magnitudes the greater has a less ratio to the same than the
less has to it ; and a magnitude has a less ratio to the less than to the greater .
Given A , B , unequal magnitudes of which A is the less ; also C , a magnitude of
the ...
8 Th . Of unequal magnitudes the greater has a less ratio to the same than the
less has to it ; and a magnitude has a less ratio to the less than to the greater .
Given A , B , unequal magnitudes of which A is the less ; also C , a magnitude of
the ...
Σελίδα 77
10 Th . That is the less magnitude of two , which has a greater ratio to a third
magnitude ; and that magnitude is the greater ... Given A to C greater than B to C ;
A is less than B. Because A and B have ratios to C , the magnitudes are of the
same ...
10 Th . That is the less magnitude of two , which has a greater ratio to a third
magnitude ; and that magnitude is the greater ... Given A to C greater than B to C ;
A is less than B. Because A and B have ratios to C , the magnitudes are of the
same ...
Σελίδα 78
ADF13 Th . If the first have to the second the same ratio which the third has to the
fourth , but the third to the fourth a greater ratio than the fifth has to the sixth ; the
first shall have to the second a greater ratio than the fifth has to the sixth . Take A
...
ADF13 Th . If the first have to the second the same ratio which the third has to the
fourth , but the third to the fourth a greater ratio than the fifth has to the sixth ; the
first shall have to the second a greater ratio than the fifth has to the sixth . Take A
...
Σελίδα 97
AA 19 Th . Similar triangles are to one another in the duplicate ratio of their
homologous sides . Given the ... because they are the consequents of the ratios (
a ) ; and the ratio of ABC to DEF is the duplicate or square o the ratio of BC to EF (
6 ) .
AA 19 Th . Similar triangles are to one another in the duplicate ratio of their
homologous sides . Given the ... because they are the consequents of the ratios (
a ) ; and the ratio of ABC to DEF is the duplicate or square o the ratio of BC to EF (
6 ) .
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Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD alternate antecedents applied Argument base bisected centre Chart chord circle circle ABC circumference common consequents Constr contained described diameter difference divided draw drawn equal angles equiangular equilateral equimultiples exceeds excess exterior extreme fore four fourth Geometry given given straight line gles greater half Hence inscribed interior join less magnitudes mean measure meet multiple namely opposite parallel parallelogram pass perpendicular plane polygon produced proportionals propositions proved Q. E. D. Recite radius ratio rectangle rectilineal figure remainders right angles School segment sides similar sine solid square straight line taken tangent third touch triangle ABC unequal Wherefore whole
Δημοφιλή αποσπάσματα
Σελίδα 90 - 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.
Σελίδα 117 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.
Σελίδα 92 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Σελίδα 79 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
Σελίδα 87 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally...
Σελίδα 26 - Triangles upon equal bases, and between the same parallels, are equal to one another.
Σελίδα 133 - If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.
Σελίδα 13 - AB be the greater, and from it cut (3. 1.) off DB equal to AC the less, and join DC ; therefore, because A in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides DB, BC are equal to the two AC, CB. each to each ; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is< equal to the triangle (4. 1.) ACB, the less to 'the greater; which is absurd.
Σελίδα 71 - If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth ; then shall...
Σελίδα 83 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is usually cited by the words
Πληροφορίες βιβλιογραφίας
| Τίτλος | Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's and Playfair's Editions ... |
| Συγγραφέας | Dennis M'Curdy |
| Εκδότης | Collins, Brother & Company, 1846 |
| Πρωτότυπο από | Πανεπιστήμιο Χάρβαρντ |
| Ψηφιοποιήθηκε στις | 1 Μαρ. 2007 |
| Μέγεθος | 138 σελίδες |
| Εξαγωγή αναφοράς | BiBTeX EndNote RefMan |