The First Six Books with NotesR. Milliken, 1822 - 179 pages |
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... proportional quantities given by Euclid , it has been asserted , that there is no other prin- ciple from which the doctrine can be demonstrated ; but that defence , it is hoped , must now be aban- doned , as the demonstrations here ...
... proportional quantities given by Euclid , it has been asserted , that there is no other prin- ciple from which the doctrine can be demonstrated ; but that defence , it is hoped , must now be aban- doned , as the demonstrations here ...
Page 91
... proportional ( A to B as B to C ) the first is said to have the third ( A to C ) a duplicate ratio of that which it ... proportional , the second is to the first as the fourth to the third . Prop . 20 . 16. By composition , when we infer ...
... proportional ( A to B as B to C ) the first is said to have the third ( A to C ) a duplicate ratio of that which it ... proportional , the second is to the first as the fourth to the third . Prop . 20 . 16. By composition , when we infer ...
Page 102
... proportional , the first to the second as the third to the fourth ( A to CD as B to EF ) and a submultiple ( a ) of the first be a submultiple of the second , an equi - submultiple ( b ) of the third is a sub- multiple of the fourth ...
... proportional , the first to the second as the third to the fourth ( A to CD as B to EF ) and a submultiple ( a ) of the first be a submultiple of the second , an equi - submultiple ( b ) of the third is a sub- multiple of the fourth ...
Page 105
... proportional ( A to B as Fig . 22 . CD to E ) they are also proportional when taken in- versely ( B to A as E to CD ) . If there be taken any equi - submultiples b and e of B and E , b is contained in A as often as e is con- tained in ...
... proportional ( A to B as Fig . 22 . CD to E ) they are also proportional when taken in- versely ( B to A as E to CD ) . If there be taken any equi - submultiples b and e of B and E , b is contained in A as often as e is con- tained in ...
Page 106
... proportional ( A to B as C to D ) the first is to the sum of the first and second as the third to the sum of the third and fourth ( A to sum of A and B as C to sum of C and D ) . Take a and c equi - submultiples of A and C , and a ( 1 ) ...
... proportional ( A to B as C to D ) the first is to the sum of the first and second as the third to the sum of the third and fourth ( A to sum of A and B as C to sum of C and D ) . Take a and c equi - submultiples of A and C , and a ( 1 ) ...
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Common terms and phrases
absurd AC and BD AC and CB AC are equal alternate angles angle ABC angle ACD angle BAC angle equal base bisected centre circumference CKMB Constr constructed contained in CD demonstrated double the rectangle double the square equal angles equal sides equal to AC equal to double equi equi-multiples equi-submultiples equiangular Euclid evident fore four magnitudes proportional four right angles given angle given circle given line given right line given triangle half a right Hypoth inscribed less line CD manner mean proportional multiple oftener parallel parallelogram perpendicular point of contact PROB produced Prop proposition radius rectangle under AC right line AB Schol segment side AC similar squares of AC submultiple taken tangent THEOR third tiple triangle ABC vertex
Popular passages
Page 145 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Page 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another : 16.
Page 28 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 22 - 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.
Page 118 - A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.
Page 146 - A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
Page 168 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...
Page 3 - 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. 4. Things which coincide with one another are equal to one another.
Page 25 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 3 - A rhombus is that which has all its sides equal, but its angles are not right angles.