Books 3-9The University Press, 1908 - Mathematics, Greek |
From inside the book
Results 1-5 of 51
Page 2
... theorem . I think however that Euclid would have maintained that it is a definition in the proper sense of the term ; and certainly it satisfies Aristotle's requirement that a " definitional statement " ( opioTikòs λóyos ) should not ...
... theorem . I think however that Euclid would have maintained that it is a definition in the proper sense of the term ; and certainly it satisfies Aristotle's requirement that a " definitional statement " ( opioTikòs λóyos ) should not ...
Page 3
... theorem that circles touching at one point do not intersect anywhere else , he has given us , before reaching that ... theorems in Book III .; 111. 8 is required for the second proof of 111. 9 which Simson selected in preference to the ...
... theorem that circles touching at one point do not intersect anywhere else , he has given us , before reaching that ... theorems in Book III .; 111. 8 is required for the second proof of 111. 9 which Simson selected in preference to the ...
Page 4
... theorem of 1. 5 was , in the text - books immediately preceding Euclid , proved by means of the equality of the two " angles of " any one segment . This latter property must therefore have been regarded as more elementary ( for whatever ...
... theorem of 1. 5 was , in the text - books immediately preceding Euclid , proved by means of the equality of the two " angles of " any one segment . This latter property must therefore have been regarded as more elementary ( for whatever ...
Page 5
... of form is meant . He adds that the definition is a theorem , or would be if " similar " had taken its final meaning . BOOK III . PROPOSITIONS . PROPOSITION I. To find the III . DEFF . 10 , 11 ] NOTES ON DEFINITIONS 5-11 5.
... of form is meant . He adds that the definition is a theorem , or would be if " similar " had taken its final meaning . BOOK III . PROPOSITIONS . PROPOSITION I. To find the III . DEFF . 10 , 11 ] NOTES ON DEFINITIONS 5-11 5.
Page 7
... theorem is a direct consequence of the theorem that , if P is any point equidistant from A and B , then P lies on the straight line bisecting AB perpendicularly . We then take any two chords AB , AC of the given circle and draw DO , EO ...
... theorem is a direct consequence of the theorem that , if P is any point equidistant from A and B , then P lies on the straight line bisecting AB perpendicularly . We then take any two chords AB , AC of the given circle and draw DO , EO ...
Common terms and phrases
ABCD angle ABC angle BAC antecedent Aristotle base bisected centre circle ABC circumference construction continued proportion corresponding sides cube number definition diameter drawn enunciation equal angles equiangular equimultiples Euclid Eutocius ex aequali four magnitudes geometrical geometrical progression given circle given straight line greater ratio greatest common measure Heiberg hypothesis Iamblichus joined less mean proportional numbers measures the number multiple multitude Nicomachus odd number parallel parallelogram pentagon polygon Porism prime number Proclus Prop proper fraction proposition PROPOSITION 13 proved rect rectangle rectangle contained rectilineal figure reductio ad absurdum remaining angle right angles segment semicircle similar and similarly similar plane numbers Simson solid numbers square number subtracted taken Theon Theon of Smyrna theorem touches the circle triangle ABC unit VIII δὲ καὶ πρὸς τὸ τοῦ
Popular passages
Page 34 - EQUAL straight lines in a circle are equally distant from the centre ; and those which are equally distant from the centre, are equal to one another.
Page 65 - 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 37 - THE straight line drawn at right angles to the diameter of a circle, from the extremity of...
Page 234 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Page 64 - From this it is manifest that if one angle of a triangle be equal to the other two it is a right angle, because the angle adjacent to it is equal to the same two ; (i.
Page 209 - 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.
Page 90 - EF at right angles (9. 1.) to AB, AC ; DF, EF produced meet one another : for, if they do not meet, they are parallel, wherefore AB, AC, which are at right angles to them, are parallel ; which is absurd : let them meet in F, and join FA ; also, if the point F be not in BC, join BF, CF : then, because AD is equal to DB, and DF common, and at right angles to AB, the base AF is equal (4.
Page 80 - In a given circle to place a straight line, equal to a given straight line which is not greater than the diameter of the circle. Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle.
Page 212 - ... be parallel to the remaining side of the triangle. Let DE be drawn parallel to BC, one of the sides of the triangle ABC : BD is to DA, as CE to EA. Join BE, CD ; Then the triangle BDE is equal to the triangle CDE*, * «.i.
Page 95 - In the same manner, it may be demonstrated that the straight lines EC, ED, are each of them equal to EA or EB ; therefore the four straight lines EA, EB, EC, ED, are equal to one another ; and the circle described from the centre E, at the distance of one of them, shall pass through the extremities of the other three, and be described about the square ABCD.