The rudiments of mathematics; designed for the use of students at the universities: containing an introduction to algebra, remarks on the first six books of Euclid, the elements of plane trigonometry - Softcover

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1809 Excerpt: ...not see that this proposition (the converse of the former) is by no means true? May not a triangle and a square be equal figures, that is, have equal areas? And can a right-lined figure, having only three sides, laid upon another having four sides, agree with it every where in the boundary lines? Again, if two triangles have their sides respectively equal, their angles will also be respectively equal, by the 8th I. El. par. 39. But if two triangles have their angles respectively equal, it does not follow that their sides will be respectively equal;_ this may or may not be true, as it happens, as will be shewn hereafter. Converse propositions, therefore, need a proof, notwithstanding this has been termed superfluous and impertinent by some who call themselves mathematicians. ON THE DEFINITIONS. 16. In a definition, all those ideas are enumerated which constitute the abstract idea of the figure. Thus a triangle is a right-lined figure having three angles (using the word figure in Euclid's sense, definition 14). Now this circumstance necessarily implies many others: for instance, that the sum of the three angles makes two right angles: this property is included in, and is virtually affirmed in, the definition; and thus all the propositions in Euclid are virtually included and contained in the definitions and axioms. 17-We said, that a definition was an enumeration of all the particular ideas that constitute the whole complex idea. It hence follows, that. a simple idea cannot be defined. Another word may be used; he who does not know what the word red means, may understand the word rouge, but this is not laying down a definition. So the attempt to define a straight line (see Euclid, Def. 4.) is absurd. It is full as difficult to understand what is meant by a &q...