## Euclid's Elements of Geometry: From the Latin Translation of Commandine. To which is Added, a Treatise of the Nature and Arithmetic of Logarithms ; Likewise Another of the Elements of Plain and Spherical Trigonometry : with a Preface ... |

### From inside the book

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Page 117

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**Ratio**, is a certain mutual Habitude of the Magni- tudes of the same kind , according to Quantity . IV . Magnitudes are said to have Proportion to each other , which being multiplied can exceed one another , V. Magnitudes are said to be ... Page 118

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**Ratio**, the first to the second , as the third to the fourth . VI . Magnitudes that have the same Proportion , are ...**Ratio**, according to the Conditions that Magnitudes in the same**Ratio**must have , laid down ' in the fifth ... Page 120

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**Ratio**to what - it has to the Second . XI . But when four Magnitudes are Proportionals , the First shall have a Trilicate**Ratio**to the Fourth of what it has to the Second ; and fo always one more in Order , as the Proportionals fhall be ... Page 130

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**Ratio**to D , than C has to D. I say , moreover , that D has a greater**Ratio**to C , than it has to AB ; for the fame Construction remaining , we demonstrate , as be- fore , that N exceeds K , but not FH . And N is a Multiple of D ... Page 140

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**Ratio**. The Demonstration of Converse**Ratio**, laid down in this Corollary , is only particular . For Alternation ( which is used herein ) cannot be apply'd but when the four proportional Magnitudes are all of the same Kind , as will ...### Common terms and phrases

alfo alſo equal Altitude Angle ABC Angle BAC Angle BCD Bafe Baſe BC becauſe biſected Center Circle ABCD Circle EFGH Circumference Cofine Cone confequently contain'd Coroll Cylinder defcrib'd DEFH demonftrated deſcribed Diameter Diſtance drawn thro equal Angles equiangular equilateral Equimultiples fame Multiple fame Reaſon fimilar fince firſt folid Parallelepipedon fore fubtending given Right Line greater join leffer leſs likewife Logarithm Magnitudes Meaſure Number oppofite parallel Parallelogram perpendicular Plane Polygon Priſms Prop PROPOSITION Pyramid Quadrant Ratio Rectangle Rectangle contained remaining Angle Right Angles Right Line AC Right-lin'd Figure ſame ſay ſecond Segment Semicircle ſhall be equal Sine ſome Sphere ſtand Subtangent THEOREM theſe thoſe Triangle ABC Unity Vertex the Point Wherefore whole

### Popular passages

Page 192 - 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 162 - IF two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals : the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

Page 63 - DBA ; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater (16.

Page 154 - ... therefore the angle DFG is equal to the angle DFE, and the angle at G to the angle at E : but the angle DFG is equal to the angle ACB...

Page 100 - About a given circle to describe a triangle equiangular to a given triangle. Let ABC be the given circle, and DEF the given triangle; it is required to describe a triangle about the circle ABC equiangular to the triangle DEF.

Page 17 - CF, and the triangle AEB to the triangle CEF, and the remaining angles to the remaining angles, each to each, to which...

Page 212 - CD; therefore AC is a parallelogram. In like manner, it may be proved that each of the figures CE, FG, GB, BF, AE, is a parallelogram...

Page 233 - If two right-angled triangles have their hypotenuses equal, and one side of the one equal to one side of the other, the triangles are congruent.

Page 166 - ABG ; (vi. 1.) therefore the triangle ABC has to the triangle ABG the duplicate ratio of that which BC has to EF: but the triangle ABG is equal to the triangle DEF; therefore also the triangle ABC has to the triangle DEF the duplicate ratio of that which BC has to EF. Therefore similar triangles, &c.

Page 95 - If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.