_Duplication of the Cube._--In his "Young Geometrician; or, Practical Geometry without Compasses," 1865, Mr. Oliver Byrne's 40th Problem is as follows:-- Let AB be the side of a given cube BD. It is required to find AC, the side of another cube CE, so that the solid contents of the cube CE are double the solid contents of the cube BD. Ancient and modern mathematicians (says Mr. Byrne) have in vain attempted to solve this problem geometrically, that is, by the ruler and compasses only. Let AB = BG = GR = RQ = QP = QO = OR = VZ. The length of the shortest side of the lesser set square; a line of any other given length may be applied. Draw OP and VR parallel to it; then apply the set squares in close contact, the edge OV of OVT passing through the point O, while the points of V and Z of ZSV fall exactly on the lines RV, RZ. Then draw the line ZBC, cutting FA produced in C; then the cube on AC is double the cube on AB. PLATE V. _Trisection of an Angle._--In his work entitled _Young Geometrician_, 1865, Mr. Oliver Byrne gives as the 39th Problem: To divide a given angle BAC into three equal angles:-- The line A _m_ is made = _p q_, the least side of the lesser triangular ruler; by (II) _p m_ is drawn parallel, and _m n_ perpendicular to AB. Then both rulers are kept in motion, and at the same time in close contact, as represented in the figure, until _p_ falls on the line _p m_, and _n_ on the line _m n_; _r n_A passing through the angular point A. Then the angle DAB is one-third of the angle CAB. Mr. Byrne asserts that this problem is not capable of solution by the straight line and circle. Mathematicians have in vain attempted to solve it geometrically, that is, by the ruler and compasses only. PLATE VI.--FIGURE 1. _Perpetuum Mobile._ Desaguliers demonstrated the absurdity of attempting to raise weights enclosed in a cellular wheel, simply by providing for their approach in succession nearer to the centre on the ascending side, while they should be projected further from the centre on the descending side. He remarks:-- Those who think the velocity of the weight is the line it describes, expect that that weight shall be overpoised, which describes the shortest line, and therefore contrive machines to cause the ascending weight to describe a shorter line than the descending weight. For example, in the circle A B D _a_, the weights A and B being supposed equal, it is imagined that, if by any contrivance whatever, whilst the weight A describes the arc A _a_, the weight B is carried in any arc, as B _b_, so as to come nearer the centre in its rising, than if it went up the arc B D; the said weight shall be overpoised, and consequently, by a number of such weights, a perpetual motion produced.