The Elements of Euclid: In which the Propositions are Demonstrated in a New and Shorter Manner Than in Former Translations, and the Arrangement of Many of Them Altered, to which are Annexed Plain and Spherical Trigonometry, Tables of Logarithms from 1 to 10,000, and Tables of Sines, Tangents, and Secants, Natural and Artificial |
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Page 65
... fame kind , according to quantity . IV . Magnitudes have proportion to each other ; which , being multi- plied , can ... multiple of the firft be equal to the multiple of the second , then the multiple of the third will be equal to the ...
... fame kind , according to quantity . IV . Magnitudes have proportion to each other ; which , being multi- plied , can ... multiple of the firft be equal to the multiple of the second , then the multiple of the third will be equal to the ...
Page 67
... multiple of the firft exceeds the multiple of the fecond , but the multiple ... fame ratio . XIII . Alternate ratio is the comparing the antecedent with the ... fame ratio ; it fhall be in the firft order of magnitudes , as the first is ...
... multiple of the firft exceeds the multiple of the fecond , but the multiple ... fame ratio . XIII . Alternate ratio is the comparing the antecedent with the ... fame ratio ; it fhall be in the firft order of magnitudes , as the first is ...
Page 68
... fame ratio ; in the first order of magnitudes , as the antecedent is to the confequent ; fo , in the fecond order of magnitudes , is the antecedent to the confequent ; and , as in the first order ... fame multiple of Let 68 THE ELEMENTS.
... fame ratio ; in the first order of magnitudes , as the antecedent is to the confequent ; fo , in the fecond order of magnitudes , is the antecedent to the confequent ; and , as in the first order ... fame multiple of Let 68 THE ELEMENTS.
Page 69
... fame multiple of the fecond C , that Book V. the third DE is of the fourth F ; and let the fifth BG be the fame multiple of the fecond C , that the fixth EH is of the fourth F ; then AG will be the fame multiple of C that DH is of F ...
... fame multiple of the fecond C , that Book V. the third DE is of the fourth F ; and let the fifth BG be the fame multiple of the fecond C , that the fixth EH is of the fourth F ; then AG will be the fame multiple of C that DH is of F ...
Page 70
... fame multiple of another magnitude , that a part taken from the one is of a part taken from the other ? then the refidue of the one shall be the fame multiple of the refidue of the other that the whole is of the whole . Let AB be the ...
... fame multiple of another magnitude , that a part taken from the one is of a part taken from the other ? then the refidue of the one shall be the fame multiple of the refidue of the other that the whole is of the whole . Let AB be the ...
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Common terms and phrases
ABCM angle ABC angle BAC arch bafe bafe BC baſe becauſe bifected Book XI circle ABCD circle EFGH circumference cofine common fection cone contained cylinder defcribe DEFH diameter draw drawn equal angles equal to AC equiangular equilateral equimultiples fame altitude fame multiple fame plain fame proportion fame reaſon fecond fegment femicircle fhall fides fimilar folid angle folid parallelopipedons fome fore fphere fquare of AC fubtending given right line greater infcribed join lefs leſs likewife magnitudes parallel parallelogram perpendicular plain paffing polygon polyhedron prifms PROP pyramid rectangle right angles right line AB right lined figure Secant Sine Tang tangent thefe THEOR theſe triangle ABC triplicate ratio Wherefore whofe bafe whoſe
Popular passages
Page 80 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 72 - F, equal to them in number, be taken two and two in the fame ratio, and if their analogy be perturbate, viz. as A is to B, fo is E to F, and B to C as D to E ; and if the firft A be greater than the third C, then the fourth D will be greater than the fixth F ; if equal, equal ; and, if lefs, lefs.
Page 91 - BAC was proved to be equal to ACD : Therefore the whole angle ACE is equal to the two angles ABC, BAC...
Page x - If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles.
Page 54 - Let ABC be the given circle, and D the given straight line, not greater than the diameter of the circle. It is required to place in the circle ABC a straight line equal to D.
Page 9 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz.
Page 13 - 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 69 - Let AB be the fame multiple of C, that DE is of F : C is to F, as AB to DE. Becaufe AB is the fame multiple of C that DE is of F ; there are as many magnitudes in AB equal to C, as there are in...
Page 91 - BGC: for the same reason, whatever multiple the circumference EN is of the circumference EF, the same multiple is the angle EHN of the angle EHF: and if the circumference BL be equal to the circumference EN, the angle BGL is also equal to the angle EHN ; (in.
Page 80 - ... reciprocally proportional, are equal to one another. Let AB, BC be equal parallelograms which have the angles at B equal, and let the sides DB, BE be placed in the same straight line ; wherefore also FB, BG are in one straight line (2.