## Elements of Geometry: Containing the Principal Propositions in the First Six, and the Eleventh and Twelfth Books of Euclid |

### From inside the book

Page 134

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**multiple**of a lefs , when the greater is equal to a certain number of times the less . 3. Ratio is a certain mutual relation of**two**magnitudes of the**fame**kind , which arifes from confidering the quan- tity of each . 4. When four ... Page 135

... multiple any one of them is of its part , the

... multiple any one of them is of its part , the

**fame multiple**will all the former be of all the latter . E F Let any number of magnitudes AB , CD be equimulti- ples of as many others E , F , each of each ; then what- ever multiple AB ... Page 136

... multiple AB is of E , the

... multiple AB is of E , the

**fame multiple**will AB and CD together be of E and F to- Q. E. D. gether , PROP . II . THEOREM . If any number of magnitudes be multiples of the fame magnitude , and as many others be the**fame multiples**of ... Page 137

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**fame multiple**of c , as the whole DH is of F. Q. E. D. PROP . III . THEOREM . If the first of four magnitudes be the**fame multiple**of the fecond as the third is of the fourth ; and if of the firft and third there be taken ... Page 138

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**fame multiple**of в that c is of D ( by Hyp . ) , and EK is equal to A , and GL to C ( by Conft . ) , EK will be the**fame multiple**of B that GL is of D. In like manner , fince KF is equal to A , and LH to c , KF will be the fame ...### Other editions - View all

Elements of Geometry: Containing the Principal Propositions in the First Six ... Euclid,John Bonnycastle No preview available - 2016 |

### Common terms and phrases

ABCD AC is equal alfo equal alſo be equal alſo be greater altitude angle ABC angle ACB angle BAC angle CAB angle DAF bafe baſe becauſe bifect cafe centre chord circle ABC circumference Conft defcribe demonftration diagonal diameter diſtance draw EFGH equiangular equimultiples EUCLID fame manner fame multiple fame plane fame ratio fecond fection fegment fhewn fide AB fide AC fimilar fince the angles folid fome fquares of AC ftand given circle given right line infcribed interfect join the points lefs leſs Let ABC magnitudes muſt oppofite angles outward angle parallelepipedons parallelogram perpendicular polygon prifm propofition proportional Q. E. D. PROP reafon rectangle of AB rectangle of AC remaining angle right angles SCHOLIUM ſhall ſpace ſquare tangent THEOREM theſe thofe thoſe triangle ABC twice the rectangle whence

### Popular passages

Page 166 - 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 73 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

Page 215 - Lemma, if from the greater of two unequal magnitudes there be taken more than its half, and from the remainder more than its half, and so on, there shall at length remain a magnitude less than the least of the proposed magnitudes.

Page 117 - 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. Draw the straight line GAH touching the circle in the point A (III. 17), and at the point A, in the straight line AH, make the angle HAG equal to the angle DEF (I.

Page 18 - To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it. LET ab be the given straight line, which may be produced to any length both ways, and let c be a point without it. It is required to draw a straight line perpendicular to ab from the point c.

Page 249 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

Page 102 - To bisect a given arc, that is, to divide it into two equal parts. Let ADB be the given arc : it is required to bisect it.

Page i - Handbook to the First London BA Examination. Lie (Jonas). SECOND SIGHT; OR, SKETCHES FROM NORDLAND. By JONAS LIE. Translated from the Norwegian. [/» preparation. Euclid. THE ENUNCIATIONS AND COROLLARIES of the Propositions in the First Six and the Eleventh and Twelfth Books of Euclid's Elements.

Page 5 - AXIOM is a self-evident truth ; such as, — 1. Things which are equal to the same thing, are equal to each other. 2. If equals be added to equals, the sums will be equal. 3. If equals be taken from equals, the remainders will be equal. 4. If equals be added to unequals, the sums will be unequal.

Page 145 - F is greater than E; and if equal, equal; and if less, less. But F is any multiple whatever of C, and D and E are any equimultiples whatever of A and B; [Construction.