## Theoretical and Practical Graphics: An Educational Course on the Theory and Practical Applications of Descriptive Geometry and Mechanical Drawing, Prepared for Students in General Science, Engineering Or Architecture |

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actual angle appear applied assumed auxiliary axis base become called centre chapter circle circular common complete cone construction containing curve cylinder described desired determined diameter direction director distance divide draw drawn edge effect element elevation ellipse employed equal evidently extremities face figure front Geometry given gives height helix horizontal horizontal line illustrated inch inclined indicated intersection join latter length light meet meridian method moving object oblique obtained original parallel pass perpendicular perspective plane portion position practical principles problem projection radius reference relation represent respectively result rolling rotation seen shade shadow shown side similarly solution space sphere straight line student surface taken tangent tangent plane tint trace triangle trochoid true usually vertex vertical vertical plane warped

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2. lappuse - The areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles.

4. lappuse - A more extended experience and more accurate measurements would teach them that the axioms were each of them false ; and that any two lines, if produced far enough each way, would meet in two points; they would, in fact, arrive at a spherical geometry accurately representing the properties of the two-dimensional space of their experience.

104. lappuse - 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.

104. lappuse - ... and which consists, in its elementary and positive acceptation, in picturing to ourselves, clearly and easily, a large and variable collection of ideal objects, as if they were really before us. ... While it belongs to the geometry of the ancients by the character of its solutions, on the other hand it approaches the geometry of the moderns by the nature of the questions which compose it. These questions are in fact eminently remarkable for that generality which constitutes the true fundamental...

105. lappuse - The projection of a point on a plane is the foot of the perpendicular from the point to the plane. The projection of a figure upon a plane is the locus of the projections of all the points of the figure upon the plane. Thus, A'B' represents the projection of AB upon plane MN.

99. lappuse - ... make the positive print, another bath is made just like the first one, except that lampblack is substituted for the burnt umber. To obtain colored positives the black is replaced by some red, blue, or other pigment. In making the copy, the drawing to be copied is put in a photographic printing-frame, and the negative paper laid on it, and then exposed in the usual manner. In clear weather, an illumination of two minutes will suffice. After the exposure, the negative is put in water to develop...

4. lappuse - ... a space, and the axiom as to parallel lines. A more extended experience and more accurate measurements would teach them that the axioms were each of them false ; and that any two lines if produced far enough each way, would meet in two points : they would in fact arrive at a spherical geometry...

121. lappuse - Any geometrical magnitude or object is said to be revolved about a right line as an axis, when it is so moved that each of its points describes the circumference of a circle whose plane is 'perpendicular to the axis, and whose centre is in the axis. By this revolution, it is evident that the relative position of the points of the object is not changed, each point remaining at the same distance from any of the other points. Thus, if the point M, Fig. 6, be revolved about an axis DD', in the horizontal...

4. lappuse - It is interesting to consider two different ways in which, without any modification at all of our notion of space, we can arrive at a system of non-Euclidian (plane or two-dimensional) geometry ; and the doing so will, I think, throw some light on the whole question.