Self-examinations in EuclidW. P. Grant, 1829 - 188 páginas |
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Página 22
... appears , however , to have adopted the safe side of the limit of his method , by supposing that DE is not to exceed DF or DL ; although he ought to have shewn that △ DEF is > than the DEG . 64 . PROP . XXVI . What is the use of that ...
... appears , however , to have adopted the safe side of the limit of his method , by supposing that DE is not to exceed DF or DL ; although he ought to have shewn that △ DEF is > than the DEG . 64 . PROP . XXVI . What is the use of that ...
Página 26
... four L3 . ¡ . L ' A , ABD together = two Ls . Hence ( XXVIII . ) , AC || BD . In the same manner it may be shewn that AB CD . .. ACDB is a Q. E. D. From this converse Prop . it appears that the defect 26 SELF EXAMINATIONS.
... four L3 . ¡ . L ' A , ABD together = two Ls . Hence ( XXVIII . ) , AC || BD . In the same manner it may be shewn that AB CD . .. ACDB is a Q. E. D. From this converse Prop . it appears that the defect 26 SELF EXAMINATIONS.
Página 27
... appears , therefore , that there ought to have been two cases . A similar defect is in the demonstration of Prop . XL . PROP . XLI . 82. From this Proposition it is easy to see that If a and are between the same || , have their bases ...
... appears , therefore , that there ought to have been two cases . A similar defect is in the demonstration of Prop . XL . PROP . XLI . 82. From this Proposition it is easy to see that If a and are between the same || , have their bases ...
Página 41
... appears that Prop . I. demonstrates Geometrically the rule of Algebra , which inculcates that the product of two quantities is equal to the sum of the products of one of them multiplied by the several parts . PROP . II . 106 ...
... appears that Prop . I. demonstrates Geometrically the rule of Algebra , which inculcates that the product of two quantities is equal to the sum of the products of one of them multiplied by the several parts . PROP . II . 106 ...
Página 42
... appears that Euclid's IV . Prop . , Book II . , is a Geometrical proof of the Binomial Theorem , for that case in which the index is 2 . PROP . V. 109. Prove this Algebraically . Since AD AC + CD , ACCD , and DB = BC CD - : . AD × DB ...
... appears that Euclid's IV . Prop . , Book II . , is a Geometrical proof of the Binomial Theorem , for that case in which the index is 2 . PROP . V. 109. Prove this Algebraically . Since AD AC + CD , ACCD , and DB = BC CD - : . AD × DB ...
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AB² ABCD AC² AD² AE x EB AE² AG² Algebraically Altitude ANALYTICAL GEOMETRY axes axiom base BD² bisect Book XI CB² CD² centre Circumscribing Circumscribing Sphere Co-ordinate Planes compound ratio DE² definition demonstration describe diameter distance Dodecagon draw a touching drawn EF² equilateral equimultiples Euclid Euclid's Elements extremities Geometric Mean Geometrical given ³ given line given point given Sphere Hence inscribed intersection join magnitude meet OP² passing Pentagon plane ABC plane YOX polygon position produced proportionals proposition Propp Prove Q. E. D. PROP quadrilateral quantities Radius regular Decagon right angles segment shew shewn straight line subtended Surface take any point tangent Theorem touch the given Trapezium triangle vertex وو
Pasajes populares
Página 9 - If two triangles have two sides of the one equal to two sides of the...
Página 18 - Any two sides of a triangle are together greater than the third side.
Página 21 - Geometry, printed anno 1760, observes in his notes, that it ought to have been shewn, that the point F falls below the line EG. This probably Euclid omitted, as it is very easy to perceive, that DG being equal to DF, the point G is in the circumference of a circle described from the centre D at the distance DF, and must be in that part of it which is above the straight line EF, because DG falls above DF, the angle EDG being greater . than the angle EDF.
Página 71 - Ratio is the relation which one quantity bears to another in respect of magnitude, the comparison being made by considering what multiple, part, or parts, one is of the other.
Página 8 - For, if the triangle ABC be applied to DEF, so that the point A may be on D, and the straight line AB upon DE ; the point B shall coincide with the point E...
Página 9 - Two triangles are equal, when the three sides of the one are equal to the three sides of the other, each to each.
Página 20 - Of the two sides DE, DF, let DE be the side which is not greater than the other, and at the point D, in the straight line DE, make (i.
Página 49 - The perpendicular is the shortest straight line that can be drawn from a given point to a given straight line; and of others, that which is nearer to the perpendicular is less than the more remote; and two, and only two, equal straight lines can be drawn from the given point to the given straight line, one on each side of the perpendicular.
Página 24 - Two straight lines which intersect one another cannot be both parallel to the same straight line.
Página 175 - If an equilateral triangle be inscribed in a circle, and the adjacent arcs cut off by two of its sides be bisected, the line joining the points of bisection shall be trisected by the sides.