The methods of algebra, so far as expressions of the first and second degrees are concerned, apply with great facility to many large classes of questions connected with straight lines, circles, and other sections of the cone. Practical facility was gained by them, frequently at the expense of reasoning: the time came when a new Descartes showed how to return to geometrical construction with means supe rior to those of algebra, in many matters connected with practice. This was Menge, the inventor of descriptive geometry. The science of perspective and many other applications of geometry to the arts had previously required isolated methods of obtaining lines, angles, or areas, described under laws not readily admitting of the appliea tiou of algebra, and its consequence, the construction of tables. The descriptive geometry is a systematised form of the method by which a ground-plan and an elevation are made to give the form and dimensions of a building. The projections of a point upon two planes at right angles to one another being given, the position of the point itself is given. From this it is possible, knowing the projections of any solid figure upon two euch planes, to lay down on either of those planes a figure similar and equal to any plane section of the solid.
In the case where the section is a curve it is constructed by laying down a large number of consecutive contiguous points. The methods by which suoh an object is to be attained were generalised and simplified by Menge, whose ' Gdomdtrie Descriptive' (the second edition of which was published in 1820) is ono of the most elegant and lucid elementary works in existence.
The methods of descriptive geometry recalled the attention of geometers to the properties of projections in general, of which such only had been particularly noticed as could be applied in the arts of design or in the investigation of primary properties of the conic sections. From the time of Menge to the present this subject has been cultivated with a vigour which has produced most remarkable results, and promises more. Pure geometry has made no advance since the time of the Greeks which gives greater help to its means of invention than that which labours of what we must call the school of Menge have effected.