Elements of Geometry and Trigonometry Translated from the French of A.M. Legendre by David Brewster: Revised and Adapted to the Course of Mathematical Instruction in the United States |
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Page 41
... passes through the centre , and is terminated on both sides by the circumference , is called a diameter . From the definition of a circle , it follows that all the radii are equal ; that all the diameters are equal also , and each ...
... passes through the centre , and is terminated on both sides by the circumference , is called a diameter . From the definition of a circle , it follows that all the radii are equal ; that all the diameters are equal also , and each ...
Page 45
... passes through two of the points just mentioned , will necessarily pass through the third , and be perpendicular to the chord . It follows , likewise , that the perpendicular raised from the middle of a chord passes through the centre ...
... passes through two of the points just mentioned , will necessarily pass through the third , and be perpendicular to the chord . It follows , likewise , that the perpendicular raised from the middle of a chord passes through the centre ...
Page 46
... pass through the three given points A , B , C. We have now shown that one circumference can always be made to pass through three given points , not in the same straight line : we say farther , that but one can be described through them ...
... pass through the three given points A , B , C. We have now shown that one circumference can always be made to pass through three given points , not in the same straight line : we say farther , that but one can be described through them ...
Page 49
... pass through each of the two centres C and D ( Prop . VI . Sch . ) . But no more than one straight line can be drawn through two points ; hence the straight line , which passes through the centres , will bisect the chord at right angles ...
... pass through each of the two centres C and D ( Prop . VI . Sch . ) . But no more than one straight line can be drawn through two points ; hence the straight line , which passes through the centres , will bisect the chord at right angles ...
Page 50
... pass through the point A , are tangent to each other . For , they have only the point A common , and if through the point A , AE be drawn perpendicular to AD , the straight line AE will be a common tangent to all the circles ...
... pass through the point A , are tangent to each other . For , they have only the point A common , and if through the point A , AE be drawn perpendicular to AD , the straight line AE will be a common tangent to all the circles ...
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Common terms and phrases
adjacent altitude angle ACB angle BAC ar.-comp base multiplied bisect Book VII centre chord circ circumference circumscribed common cone consequently convex surface cosine Cotang cylinder diagonal diameter dicular distance divided draw drawn equal angles equally distant equations equivalent feet figure find the area formed four right angles frustum given angle given line gles greater homologous sides hypothenuse inscribed circle inscribed polygon intersection less Let ABC logarithm measured by half number of sides opposite parallelogram parallelopipedon pendicular perimeter perpen perpendicular plane MN polyedron polygon ABCDE PROBLEM proportional PROPOSITION pyramid quadrant quadrilateral quantities radii radius ratio rectangle regular polygon right angled triangle Scholium secant segment side BC similar sine slant height solid angle solid described sphere spherical polygon spherical triangle square described straight line tang tangent THEOREM triangle ABC triangular prism vertex
Popular passages
Page 241 - In every plane triangle, the sum of two sides is to their difference as the tangent of half the sum of the angles opposite those sides is to the tangent of half their difference.
Page 251 - Being on a horizontal plane, and wanting to ascertain the height of a tower, standing on the top of an inaccessible hill, there were measured, the angle of elevation of the top of the hill 40°, and of the top of the tower 51° ; then measuring in a direct line 180 feet farther from the hill, the angle of elevation of the top of the tower was 33° 45' ; required the height of the tower.
Page 109 - The perimeters of two regular polygons of the same number of sides, are to each other as their homologous sides, and their surfaces are to each other as the squares of those sides (Book IV.
Page 91 - Two similar triangles are to each other as the squares described on their homologous sides. Let ABC, DEF, be two similar triangles, having the angle A equal to D, and The angle B=E.
Page 169 - THEOREM. 7?/6 convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.
Page 41 - CIRCLE is a plane figure bounded by a curved line, all the points of which are equally distant from a point within called the centre; as the figure ADB E.
Page 155 - AK. The two solids AG, AQ, having the same base AEHD are to each other as their altitudes AB, AO ; in like manner, the two solids AQ, AK, having the same base AOLE, are to each other as their altitudes AD, AM. Hence we have the two proportions, sol.
Page 86 - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. A D A' Hyp. In triangles ABC and A'B'C', To prove AABC A A'B'C' A'B' x A'C ' Proof. Draw the altitudes BD and B'D'.
Page 282 - ... 1. To find the length of an arc of 30 degrees, the diameter being 18 feet. ' Ans. 4.712364. 2. To find the length of an arc of 12° 10', or 12£°, the diameter being 20 feet.
Page 93 - ABC : FGH : : ACD : FHI. By the same mode of reasoning, we should find ACD : FHI : : ADE : FIK; and so on, if there were more triangles. And from this series of equal ratios, we conclude that the sum of the antecedents...