...angles of every rectilinear triangle are altogether equal to two right angles. In right-angled triangles the square of the hypothenuse is equal to the squares of the remaining sides. On these truths vast sciences have been reared. What the first principles of Euclid...

...meaning of werde. The proper object of mathematics is not only to impress the memory with the fact that the square of the hypothenuse is equal to the squares of the sides that contain the right angle, but also, and in a far greater degree, to accustom the mind to...

...them'; and changed his mind only when he came to the proposition, 1,47, 'that in a right angled Triangle the square of the Hypothenuse is equal to the squares of the two other sides'. A story is only a story. But look at Newton's copy 13 of Barrow's 1655 Latin epitome...

...demonstrative reasoning when establishing arithmetical propositions as well as geometrical propositions such as "the square of the hypothenuse is equal to the squares of the other two sides," 2 whereas we use experimental reasoning to show that fire burns or "that a body at rest or in motion...

...proceeds entirely from the undeterminate meaning of words, which is corrected by juster definitions. That the square of the hypothenuse is equal to the squares of the other two sides, cannot be known, let the terms be ever so exactly defined, without a train of reasoning and enquiry....

...very strong./ That there is such a City as Rome is a Proposition of which we can doubt less than that the Square of the Hypothenuse is equal to the Squares of the two sides when the latter is demonstrated, the highest Degree of testimony leaves less doubt than Demonstration:...

...same base and between the same parallels are equal. All right angled triangles have this property that the square of the hypothenuse is equal to the squares of the base and perpendicular etc. The general logical type of such proposition is All Zs are Xs of which...

...proceeds entirely from the undeterminate meaning of words, which is corrected by juster definitions. That the square of the hypothenuse is equal to the squares of the other two sides, cannot be known, let the terms be ever so exactly' defimied, without a train of reasoning and enquiry'....

...base & between the same Parallels arc equal, & that other proposition that in a right angled Triangle the square of the Hypothenuse is equal to the squares of the two other sides. Began again to read Euclid with more attention than he had done before & went through...

...natural law is the same in all people, distinguishes between the necessary truths of speculative reason, such as that the three angles of a triangle are equal to two right angles, and the contingent matters that are the concern of practical reason, such as the repaying of debts to a creditor....