## Elements of Geometry: Containing the Principal Propositions in the First Six, and the Eleventh and Twelfth Books of Euclid. With Notes, Critical and ExplanatoryJohnson, 1803 - 279 pages |

### From inside the book

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**equimultiples**whatever of the antecedents , are , each of them , either equal to , greater , or lefs , than any**equi- multiples**whatever of their confequents . 6. Inverse ratio is , when the confequents are made the antecedents , and ... Page 135

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**equimulti**- ples of as many others E , F , each of each ; then what- ever multiple AB is of E , the fame multiple will AB and CD together , be of E and F together . For fince AB is the fame multiple of E that CD is of F ( by Hyp . ) , as ... Page 137

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**equimultiples**, thefe will also be**equi- multiples**, the one of the fecond , and the other of the fourth . K E H A B Let a the first , be the fame multiple of в the fecond , as c the third , is of D the fourth ; and let EF and GH be ... Page 138

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**equimultiples**of the first and second may be taken , and fome multiple of the third fuch , that the former fhall be greater than that of the third , but the latter not greater . F L -K D BA Let AB , BC be two unequal magnitudes , and D ... Page 139

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**equimultiples**GF , FE fuch , that they may be each greater than D ; and of D take the multiples K and L fuch , that ...**equimulti**- ples of the magnitudes AB and BC , and L is a multiple of D. And , fince GF is not less than K , and ...### Contents

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### 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 BAD angle CAB bafe baſe becauſe bifect cafe centre chord circle ABC circumference confequently Conft COROLL DABC 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 firſt folid fome fquares of AC given circle given right line infcribe interfect join the points lefs leſs Let ABC magnitudes muſt oppofite angles outward angle parallelogram perpendicular polygon prifm propofition proportional Q. E. D. PROP reaſon rectangle of AB rectangle of AE remaining angle right angles ſame SCHOLIUM ſhewn ſpace ſquare tangent THEOREM theſe triangle ABC twice the rectangle uſeful whence

### Popular passages

Page 164 - 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 71 - The radius of a circle is a right line drawn from the centre to the circumference.

Page 69 - Iff a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 205 - 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 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 239 - 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 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. 5. If equals be taken from unequals, the remainders will be unequal.

Page 133 - If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the first of the sum of all the rest.

Page 143 - 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.

Page 155 - Of four proportional quantities, the first and third are called the antecedents, and the second and fourth the consequents ; and the last is said to be a fourth proportional to the other three taken in order.