=1882.= A mathematical problem may usually be attacked by what is termed in military parlance the method of “systematic approach;” that is to say, its solution may be gradually felt for, even though the successive steps leading to that solution cannot be clearly foreseen. But a Descriptive Geometry problem must be seen through and through before it can be attempted. The entire scope of its conditions, as well as each step toward its solution, must be grasped by the imagination. It must be “taken by assault.” --CLARKE, G. S. _Quoted in W. S. Hall: Descriptive Geometry (New York, 1902), chap. 1._ =1883.= The grand use [of Descriptive Geometry] is in its application to the industrial arts;--its few abstract problems, capable of invariable solution, relating essentially to the contacts and intersections of surfaces; so that all the geometrical questions which may arise in any of the various arts of construction,--as stone-cutting, carpentry, perspective, dialing, fortification, etc.,--can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circumstances of each case. This creation must be very important in the eyes of philosophers who think that all human achievement, thus far, is only a first step toward a philosophical renovation of the labours of mankind; towards that precision and logical character which can alone ensure the future progression of all arts.... Of Descriptive Geometry, it may further be said that it usefully exercises the student’s faculty of Imagination,--of conceiving of complicated geometrical combinations in space; and that, while it belongs to the geometry of the ancients by the character of its solutions, it approaches to the geometry of the moderns by the nature of the questions which compose it.--COMTE, A. _Positive Philosophy [Martineau], Bk. 1, chap. 3._ =1884.= There is perhaps nothing which so occupies, as it were, the middle position of mathematics, as trigonometry. --HERBART, J. F. _Idee eines ABC der Anschauung; Werke (Kehrbach) (Langensalza, 1890), Bd. 1, p. 174._ =1885.= Trigonometry contains the science of continually undulating magnitude: meaning magnitude which becomes alternately greater and less, without any termination to succession of increase and decrease.... All trigonometric functions are not undulating: but it may be stated that in common algebra nothing but infinite series undulate: in trigonometry nothing but infinite series do not undulate.--DE MORGAN, A. _Trigonometry and Double Algebra (London, 1849), Bk. 1, chap. 1._