Elements of plane geometry, book i, containing nearly the same propositions as the first book of Euclid's Elements |
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Seite 15
They were careful to exclude the idea of motion from their reasonings , because they imagined that its introduction would impair the accuracy of their conclusions . Prosecuting the study of the science on this principle alone ...
They were careful to exclude the idea of motion from their reasonings , because they imagined that its introduction would impair the accuracy of their conclusions . Prosecuting the study of the science on this principle alone ...
Seite 20
By this method of introducing the principle of variation an approximation only to the truth is attained , whereas , if the same principle were applied directly , the conclusion might be arrived at with perfect accuracy .
By this method of introducing the principle of variation an approximation only to the truth is attained , whereas , if the same principle were applied directly , the conclusion might be arrived at with perfect accuracy .
Seite 21
... where it can be introduced with advantage , by leading to more satisfactory conclusions , and obviating difficulties which admit of no other mode of solution . It might also prove very useful to students to have some acquaintance ...
... where it can be introduced with advantage , by leading to more satisfactory conclusions , and obviating difficulties which admit of no other mode of solution . It might also prove very useful to students to have some acquaintance ...
Seite 22
... nothing which admitted of proof ; and every proposition is established by the most rigorous demonstration , without allowing any doubtful element to enter which might in the smallest degree impair the validity of his conclusions .
... nothing which admitted of proof ; and every proposition is established by the most rigorous demonstration , without allowing any doubtful element to enter which might in the smallest degree impair the validity of his conclusions .
Seite 24
If the conclusions are in harmony with the axiom applicable to the particular case , they are received as veritable truths ; but if they are at variance with it , they are rejected as false and absurd .
If the conclusions are in harmony with the axiom applicable to the particular case , they are received as veritable truths ; but if they are at variance with it , they are rejected as false and absurd .
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Elements of Plane Geometry, Book: Containing Nearly the Same Propositions As ... Euclid Keine Leseprobe verfügbar - 2008 |
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ABC is equal AC is equal acute adjacent alternate angles ancient angle ACD angle BAC angles ABC appears application assume axiom base BC bisect called centre circle circumference coincide common conclusion construction definition demonstration describe determined diagonal draw drawn elements employed established Euclid extended exterior angle extremities fall four right angles geometers geometry given straight line greater half Hence included angle interior opposite angle intersect introduced join knowledge less Let ABC magnitudes manner means meet method mind mode necessary obtuse parallel lines parallelogram perpendicular plane position principle problem produced proof properties PROPOSITION proved radiant reason rectangle rectilineal figure remaining respects side AB side AC surfaces THEOR thing third triangle ABC triangles are equal truths unequal vertex wherefore whole
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Seite 43 - If two triangles have two sides and the included angle of the one, equal to two sides and the included angle of the other, each to each, the two triangles will be equal.
Seite 46 - Any two angles of a triangle are together less than two right angles.
Seite 37 - The angles which one straight line makes with another upon one tide of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.
Seite 57 - Through a given point to draw a straight line parallel to a given straight line, Let A be the given point, and BC the given straight line : it is required to draw through the point A a straight line parallel to BC.
Seite 38 - ... in one and the same straight line. At the point B in the straight line AB, let the two straight lines BC, BD upon the opposite sides of AB, make the adjacent angles ABC, ABD, equal together to two right angles. BD is in the same straight line with CB.
Seite 68 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Seite 34 - LET it be granted that a straight line may be drawn from any one point to any other point.
Seite 64 - Parallelograms upon the same base, and between the same parallels, are equal to one another.
Seite 46 - If one side of a triangle be produced, the exterior angle is greater than either of the interior, and opposite angles.
Seite 34 - Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal.