## Manual of Euclid: books iv., v., vi., by J.A. Galbraith and S. Haughton |

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Annotation arc BC base bisected called centre circumscribed common consequent construct contained Corollary DEFINITION described diameter divided double draw drawn equal angles equiangular equilateral equimultiples exscribed circles extremities fifth figure fore four fourth given circle given line given point given triangle greater greater ratio harmonically inscribed intersection joining less locus magnitudes manner mean meet multiple opposite sides parallel parallelogram pass perpendicular polygon produced Proof Proof.—Because Prop proportional PROPOSITION prove Q. E. D. PROPOSITION radius ratio rectangle rectilinear figure remaining right angles right line scribed sector segments sides similar similarly sixth square Statement.-Let straight line taken tangents third triangle ABC vertex vertical Wherefore whole ανάλογον δε και λόγον προς Πρότασις τε

### Popular passages

Page 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. Let A the first be the same multiple of B the second, that C the third is of D the fourth ; and of A...

Page 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...

Page 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.

Page 70 - DE ; but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.

Page 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.

Page 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.

Page 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...

Page 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...

Page 66 - 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.

Page 68 - CF; and because it has been shewn, that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD; and as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF; therefore, as the base BC is to the base CD, so is the parallelogram EC to the parallelogram CF.