Manual of Euclid: Books IV, V, VI.Cassell, Petter, and Galpin, 1868 |
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Página 30
... proportional , the first is said to have to the third a ratio duplicate ( two - fold ) of that which it has to the second . Annotation . Thus if xyz , the ratio of x : z is duplicate of the ratio : y . This is expressed algebraically ...
... proportional , the first is said to have to the third a ratio duplicate ( two - fold ) of that which it has to the second . Annotation . Thus if xyz , the ratio of x : z is duplicate of the ratio : y . This is expressed algebraically ...
Página 40
... proportional , they will be proportional also by inversion . Algebraical Statement . — If and if Annotation . a : b :: x : y , a ' = ma , = mx ; and , b ' = nb , y ' = ny . m and n being whole numbers ; then dividing the first by the ...
... proportional , they will be proportional also by inversion . Algebraical Statement . — If and if Annotation . a : b :: x : y , a ' = ma , = mx ; and , b ' = nb , y ' = ny . m and n being whole numbers ; then dividing the first by the ...
Página 49
... proportional ; as one of the antecedents is to its consequent , so are all the antece- dents to all the consequents . Statement . - Let any number of magnitudes , A , B , C , D , E , and F , be proportional ; that is , as A is to B , so ...
... proportional ; as one of the antecedents is to its consequent , so are all the antece- dents to all the consequents . Statement . - Let any number of magnitudes , A , B , C , D , E , and F , be proportional ; that is , as A is to B , so ...
Página 54
... proportional , they will be proportional also by division . Statement . - Let AB , BE , CD , and DF , be propor- tional ; that is , as AB to BE , so is CD to DF . And let AE be the excess of AB above BE ; and CF the ex- cess of CD above ...
... proportional , they will be proportional also by division . Statement . - Let AB , BE , CD , and DF , be propor- tional ; that is , as AB to BE , so is CD to DF . And let AE be the excess of AB above BE ; and CF the ex- cess of CD above ...
Página 56
... proportional , they will also be propor- tional by composition . Statement . — Let AE be to EB as CZ to ZD ; I say that AB is to BE as CD to DZ . Construction.For , if not , A let AB be to BE as CD to DH . Then DH is either less or ...
... proportional , they will also be propor- tional by composition . Statement . — Let AE be to EB as CZ to ZD ; I say that AB is to BE as CD to DZ . Construction.For , if not , A let AB be to BE as CD to DH . Then DH is either less or ...
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Manual of Euclid: Books IV., V., VI., by J.A. Galbraith and S. Haughton Euclides Sin vista previa disponible - 2015 |
Términos y frases comunes
angle BAC antece arc BC bisected CASSELL'S centre circumscribed circle construct the triangle Corollary diameter draw drawn equal angles equiangular equilateral equimultiples exscribed circles find the locus fore four magnitudes fourth given circle given in position given line given point given ratio given triangle greater ratio harmonically homologous homologous sides inscribed intersection less multiple parallel parallelogram perpendicular polygon Proof Prop Q. E. D. Annotation Q. E. D. PROPOSITION quadrilateral radical axis radius rectangle rectilinear figure right angles right line scribed segments square Statement.-Let straight line tangents tiple triangle ABC vertex vertical angle Wherefore αἱ ἀνάλογον γωνίας δὲ δοθέντι ἐν ἔσται ἴσα ἰσάκις ἴσας ἴσον καὶ κύκλον λόγον μεγέθη Ὅρος ὅταν περὶ πρὸς τὸ Πρότασις τὰ τὰς τὴν τῆς τοῖς τὸν αὐτὸν τοῦ τρίτον τῷ τῶν ὡς
Pasajes populares
Página 28 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal...
Página 38 - If the first be the same multiple of the second which the third is of the fourth, and if of the first and third there be taken equimultiples, these shall be equimultiples, the one of the second, and the other of the fourth.
Página 3 - IN a given circle to inscribe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle ; it is required to inscribe in the circle ABC a triangle equiangular to the triangle DEF.
Página 72 - The sides about the equal angles of equiangular triangles are proportionals ; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios. Let ABC, DCE be equiangular triangles, having the angle ABC equal to the angle DCE, and the angle ACB to the angle DEC, and consequently * the angle BAC equal to the a 32.
Página 92 - CF ; but K has to M the ratio which is compounded of the ratios of the sides ; therefore also the parallelogram AC has to the parallelogram CF the ratio which is compounded of the ratios of the sides. Wherefore equiangular parallelograms, &c.
Página 76 - 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.
Página 69 - DE, and between the same parallels DE, BC; but ADE is another triangle ; and equal magnitudes have the same ratio to the same magnitude; (v. 7.) therefore, as the triangle BDE is to the triangle ADE...
Página 71 - Now let BD be to DC, as BA to AC, and join AD ; the angle CAD is equal to the angle DAE. The same construction being made, because BD is to DC as BA to AC ; and also BD to DC, BA to AF (2.
Página 29 - When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth...
Página 60 - D, and other four E, F, G, H, which, two and two, have the same ratio, viz. as A is to B, so is E to F; and as B...