Page:Carroll - Euclid and His Modern Rivals.djvu/231

. § 1.] and 'minor' angles which bear the name EGA; do. for EGB; do. for AGH; and do. for BGH. That is, eight angles at G alone. There are sixteen altogether.

P. 17, Th. 30. 'If a quadrilateral has two opposite sides equal and parallel, it is a Parallelogram.'

This re-asserts part of its own data.

P. 17, Th. 31. 'Straight Lines that are equal and parallel have equal projections on any other straight Line; conversely, parallel straight Lines that have equal projections on another straight Line are equal.'

The first clause omits the case of Lines that are equaland in one and the same straight Line. The second clause



is not true: if the parallel Lines are at right angles to the other Line, their projections are equal, both being zero, whether the Lines are equal or not.

P. 18, Th. 32. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then etc.'

The subject of this Proposition is inconceivable: there are three intercepts, and by no possibility can these three be equal.

P. 25, Prob. 5. 'To construct a rectilineal Figure equal to a given rectilineal Figure and having the number of its sides one less than that of the given Figure.'

May I ask you to furnish me with the solution of this