Elements of Geometry: Containing the First Six Books of Euclid with a Supplement on the Quadrature of the Circle, and the Geometry of Solids : to which are Added, Elements of Plane and Spherical Trigonometry |
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Page 107
... fourth , equal to it , or less ; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth . 6. Magnitudes are said to be proportionals , when the first has the same ratio to the ...
... fourth , equal to it , or less ; then the first of the magnitudes is said to have to the second the same ratio that the third has to the fourth . 6. Magnitudes are said to be proportionals , when the first has the same ratio to the ...
Page 108
... fourth ; or that the first is to the third as the second to the fourth : See Prop . 16 . of this Book . 15. Invertendo , by inversion : When there are four proportionals , and it is inferred , that the second is to the first , as the fourth ...
... fourth ; or that the first is to the third as the second to the fourth : See Prop . 16 . of this Book . 15. Invertendo , by inversion : When there are four proportionals , and it is inferred , that the second is to the first , as the fourth ...
Page 109
... fourth of the first rank , so is the third from the last , to the last but two , of the second rank ; and so on in a cross , or inverse , order ; and the inference is as in the 19th definition . It is demonstrated in the 23d Prop . of ...
... fourth of the first rank , so is the third from the last , to the last but two , of the second rank ; and so on in a cross , or inverse , order ; and the inference is as in the 19th definition . It is demonstrated in the 23d Prop . of ...
Page 111
... fourth , and if any equimultiples whatever be taken of the first and third , and any whatever of the second and fourth ; the multiple of the first shall have the same ratio to the multiple of the second , that the multiple of the third ...
... fourth , and if any equimultiples whatever be taken of the first and third , and any whatever of the second and fourth ; the multiple of the first shall have the same ratio to the multiple of the second , that the multiple of the third ...
Page 112
... fourth ; the first is to the second as the third to the fourth . First , if mA , mB be equimultiples of the magnitudes A and B , mA : A : mB : B. Take of mA and B equimultiples by any number n ; and of A and B equimultiples by any ...
... fourth ; the first is to the second as the third to the fourth . First , if mA , mB be equimultiples of the magnitudes A and B , mA : A : mB : B. Take of mA and B equimultiples by any number n ; and of A and B equimultiples by any ...
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Common terms and phrases
ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC base BC bisected centre chord circle ABC circumference cosine cylinder definition demonstrated described diameter divided draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given rectilineal given straight line greater Hence hypotenuse inscribed join less Let ABC Let the straight magnitudes meet opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROB PROP proportional proposition quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle solid parallelopiped spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore
Popular passages
Page 51 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Page 29 - Straight lines which are parallel to the same straight line are parallel to one another. Triangles and Rectilinear Figures. The sum of the angles of a triangle is equal to two right angles.
Page 12 - To draw a straight line through a given point parallel to a given straight line. Let A be the given point, and BC the given straight line.
Page 11 - Let it be granted that a straight line may be drawn from any one point to any other point. 2. That a terminated straight line may be produced to any length in a straight line. 3. And that a circle may be described from any centre, at any distance from that centre.
Page 72 - To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle. First, let A be a given point without the given circle BCD : it is required to draw a straight line from A which shall touch the circle.
Page 84 - 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.
Page 80 - The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle.
Page 22 - Any two sides of a triangle are together greater than the third side.
Page 53 - If 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 35 - Parallelograms upon the same base and between the same parallels, are equal to one another.