down into the mildest proportions when superfluities areomitted, and only so much of it proved as is really necessary to include the whole. Euclid proves all that is valuable in it in the course of I. 29, and I do not see any sufficient reason for stating and proving it as a separate Theorem.
Th. 23, Cor. (p. 46) is the rest of Euc. I. 29: old proof.
Th. 24 (p. 46) is Euc. I. 30, proved as a Contranominal of Playfair's Axiom.
Th. 25, 26, and Cor. (pp. 47, 48) are Euc. I. 32 and Corollaries: old proof.
Th. 27, 1st part (p. 50), is a needless repetition of part of the Corollary to Th. 23.
Th. 27, 2nd part (p. 50), is part of Euc. I. 34: old proof.
Th. 28 (p. 51) is the rest of Euc. I. 34, proved as in Euclid.
Th. 29 (p. 52) is new. 'If two Parallelograms have two adjacent sides of the one respectively equal to two adjacent sides of the other, and likewise an angle of the one equal to an angle of the other; the Parallelograms are identically equal.' This might be a useful exercise to set; but really it does not seem of sufficient importance to be selected for a Manual.
Th. 30 (p. 53) is Euc. I. 33,: old proof.
Th. 31 (p. 54) is new. 'Straight Lines which are equal and parallel have equal projections on any other straight Line; conversely, parallel straight Lines which have equal projections on another straight Line are equal; and equal straight Lines, which have equal projections on another straight Line, are equally inclined to that Line.' The first and third
