"Since, however, the breadth of the faintest line is so slight as to be incapable of measurement, except by an instrument under the microscope, it follows that the assumption that there can be lines without breadth is so nearly true that our senses, when unassisted by art, can not detect the error. Formerly, and until the invention of the micrometer, in the seventeenth century, it was impossible to detect it at all. Hence, the conclusions of the geometrician approximate so closely to truth that we are justified in accepting them as true. The flaw is too minute to be perceived. But that there is a flaw appears to me certain. It appears certain that, whenever something is kept back in the premises, something must be wanting in the conclusion. In all such cases, the field of inquiry has not been entirely covered; and part of the preliminary facts being suppressed, it must, I think, be admitted that complete truth be unattainable, and that no problem in geometry has been exhaustively solved."[5] The fallacy which underlies Mr. Buckle's contention is thus clearly exposed by the author of "The Natural History of Hell." "If it be conceded that lines have breadth, then all we have to do is to assign some definite breadth to each line--say the one-thousandth of an inch--and allow for it. But the lines of the geometer have no breadth. All the micrometers of which Mr. Buckle speaks depend, either directly or indirectly, upon lines for their graduations, and the positions of these lines are indicated by rulings or scratches. Now, in even the finest of these rulings, as, for example, those of Nobert or Fasoldt, where the ruling or scratching, together with its accompanying space, amounts to no more than the one hundred and fifty thousandth part of an inch, the scratch has a perceptible breadth. But this broad scratch is not the line recognized by the microscopist, to say nothing of the geometer. The true line is a line which lies in the very center of this scratch and it is certain that this central line has absolutely no breadth at all."[6] It must be very evident that if Mr. Buckle's contention that geometrical lines have breadth were true, then some of the fundamental axioms of geometry must be false. It could no longer hold true that "the whole is equal to all its parts taken together," for if we divide a square or a circle into two parts by means of a line which has breadth, the two parts cannot be equal to the whole as it formerly was. As a matter of fact, Mr. Buckle's lines are saw-cuts, not geometrical lines. Geometrical points, lines, and surfaces, have no material existence and can have none. An ideal conception and a material existence are two very different things.