## Lectures on the Principles of Demonstrative Mathematics |

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admit adopted affections algebra ancients angles appears applied argument Aristotle arithmetical assume assumption axiom Barrow circle coincide Compare comparison conceive conception conclusions congruity consequence consistency correct defined definition demonstration dependent derived difficulty direct doctrine Elements equal establish Euclid evidence example existence express extension fact fall figure former four fourth geometry given gives greater idea important impossible instance kind knowledge latter lecture less limits magnitudes mathematical means merely method mind multiple nature necessity notation notion objection once operations origin parallels particular philosophers plane Plato position possess present principles proceed Proclus produced Prop proportion proportionality proposition proved quantities ratio reason reference relation remark require right angles senses side simple space square straight line symbols taken term thing third tion triangle true truth universal whilst writers

### Popular passages

Page 64 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

Page 38 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

Page 52 - Any two sides of a triangle are together greater than the third side.

Page 96 - ... 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: or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth...

Page 122 - Whatever form is algebraically equivalent to another when expressed in general symbols, must continue to be equivalent whatever those symbols denote.

Page 17 - It is certain that from its completeness, uniformity and faultlessness, from its arrangement and progressive character, and from the universal adoption of the completest and best line of argument, Euclid's " Elements " stand preeminently at the head of all human productions.

Page 38 - Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles.

Page 67 - Parallel straight lines are such as are in the same plane, and which being produced ever so far both ways, do not meet.

Page 88 - But when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth ; and if equal, equal; if less, less; (v.

Page 25 - That all our cognition," he says, " begins with experience, there is not any doubt ; for how otherwise should the faculty of cognition be awakened into exercise, if this did not occur through objects which affect our senses...