The proportions of Gothic buildings were well studied, their forms were always lofty, their gables sharp, and their general composition more or less pyramidal. Remarkable numerical relations between the dimensions of the different parts of a great Gothic cathedral can be discovered upon careful examination in most, if not all instances, and there can be little doubt that a system of geometrical proportions ran through the earlier design, and that much of the harmony and beauty which the buildings present is traceable to this fact. Independent of this the skill with which subordinate features and important ones are fitted to their respective positions, both by their dimensions and by their relative elaboration or plainness, forms a complete system of proportion, making use of the word in its broadest sense; and the results are extremely happy. Apparent size was imparted to almost every Gothic building by the smallness, great number, and variety of its features, and by the small size of the stones employed. The effect of strength is generally, though not perhaps so uniformly, also obtained, and dignity, beauty, and harmony are rarely wanting. Symmetry, though not altogether overlooked, has but a slender hold upon Gothic architects. It is far more observed in the interior than in the exterior of the buildings; but it must be remembered that symmetry formed the basis of many designs which, owing to the execution having been carried on through a long series of years and by different hands, came to be varied from the original intentions. Thus, for example, Chartres is a cathedral with two western towers. One of these was carried up and its spire completed in the twelfth century. The companion spire was not added till the end of the fifteenth, when men's ideas as to the proportions, shape, ornaments, and details of a spire had altered entirely;--the later architect did not value symmetry enough to think himself bound to adhere either to the design or to the height of the earlier spire, so we have in this great façade two similar flanking towers but spires entirely unlike. What happened at Chartres happened elsewhere. The original design of buildings was in the main symmetrical, but it was never considered that symmetry was a matter so important as to require that much sacrifice should be made to preserve it.