An Elementary Treatise on the Geometrical and Algebraical Investigation of Maxima and Minima: Being the Substance of a Course of Lectures Delivered Conformably to the Will of Lady Sadler : to which is Added A Selection of Propositions Deducible from Euclid's ElementsJ. Smith, 1817 - 436 páginas |
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Página 72
... ratios be each greater than a given ratio , the ratio of the sum of their antecedents , to the sum of their consequents , shall be greater than that given ratio . PROP . XIII . ( 11. ) If the first of four magnitudes have a greater ratio ...
... ratios be each greater than a given ratio , the ratio of the sum of their antecedents , to the sum of their consequents , shall be greater than that given ratio . PROP . XIII . ( 11. ) If the first of four magnitudes have a greater ratio ...
Página 73
... greater ratio to the second than the third has to the fourth , the first shall have to the third a greater ratio than the second has to the fourth . PROP . XVII . ( IV . ) If the first , together with the second , of four magnitudes ...
... greater ratio to the second than the third has to the fourth , the first shall have to the third a greater ratio than the second has to the fourth . PROP . XVII . ( IV . ) If the first , together with the second , of four magnitudes ...
Página 74
... greater ratio than the third , together with the fourth , has to the fourth . ( VI . ) If the first term of a ratio be less than the second , the ratio shall be increased by adding the same quantity to both terms ; but if the first term be ...
... greater ratio than the third , together with the fourth , has to the fourth . ( VI . ) If the first term of a ratio be less than the second , the ratio shall be increased by adding the same quantity to both terms ; but if the first term be ...
Página 75
... greater ratio , than the first , together with the second , has to the third , together with the fourth . PROP . XIX . ( x . ) If any number of magnitudes be proportionals , their differences shall , also , be proportionals . • PROP ...
... greater ratio , than the first , together with the second , has to the third , together with the fourth . PROP . XIX . ( x . ) If any number of magnitudes be proportionals , their differences shall , also , be proportionals . • PROP ...
Página 76
... greater ratio than the second has to the third , in the latter ; and if , also , the second have to the third , in the former set , a greater ratio than the first has to the second , in the latter ; then shall the first have to the ...
... greater ratio than the second has to the third , in the latter ; and if , also , the second have to the third , in the former set , a greater ratio than the first has to the second , in the latter ; then shall the first have to the ...
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Términos y frases comunes
ABCD aggregate Algebra altitude angle equal apparent magnitude base Binomial Theorem bisected chord circumference coefficients Cx² derivative describe a circle diameter divided double draw a straight equal perimeter equation equilateral triangle Euclid Euclid's Elements find a point finite straight line function Geometry given angle given circle given finite straight given point given ratio given straight line given triangle greater ratio greatest hypotenuse inscribed isosceles triangle less Let ABC lines be drawn magnitudes MAXIMA AND MINIMA maximum minimum number of sides parallel parallelepiped parallelogram plane prism produced PROP proposition quadrilateral rectilineal figure rectangle contained regular polygon rhomb right angles right-angled triangle scalene triangle SCHOLIUM sector segment shewn square straight line drawn straight line joining subtended tangent Theorem trapezium triangle ABC variable quantity velocity vertex vertical angle wherefore
Pasajes populares
Página 81 - 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.
Página 83 - 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. D c A' D' Hyp. In triangles ABC and A'B'C', ZA = ZA'. To prove AABC = ABxAC. A A'B'C' A'B'xA'C' Proof. Draw the altitudes BD and B'D'.
Página 56 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square...
Página 36 - In every triangle, the square of the side subtending any of the acute angles is less than the squares of the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle. Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC, one of the sides containing it, let fall the perpendicular...
Página 32 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.
Página 85 - Divide a straight line into two parts such that the rectangle contained by the whole line and one of the parts shall be equal to the square on the other part.
Página 12 - If two triangles have the three angles of one equal to the three angles of the other, each to each, do you think the two triangles are necessarily alike in every respect ? 5.
Página 16 - 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.
Página 154 - 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.
Página 22 - If from the right angle of a right-angled triangle, two straight lines be drawn, one perpendicular to the base, and the other bisecting it, they will contain an angle equal to the difference of the two acute angles of the triangle.