CHAPTER XI.

HERBERT SPENCER (CONTINUED).

The Analysis of Reason-The Fundamental Intuition-The Contrasted Theories of Perception.

IN the second volume of Spencer's "Principles of Psychology," the author apologizes for the abstruseness of the opening portions of the work, and explains that the method which he adopts, namely, that of a systematic analysis, requires that it should begin with the most complex and special forms of intellectual activity, and progress in stages to the simplest or most general. He further says that this method will tax the powers of even the habitual student; and to those who are unaccustomed to introspection (or the study of the operations of the mind) he recommends patience, and holds out the reward of an ultimate comprehension of the subject if they will but persevere.

The first words of the second chapter are these: "Of intellectual acts, the highest are those which constitute Conscious Reasoning-[or] called conscious to distinguish it from the unconscious or automatic reasoning that forms so large an element in ordinary perception. Of conscious reasoning, the kind containing the greatest number of components definitely combined is Quantitative Reasoning. And of this, again, there is a division, more highly involved than the rest, which we may class apart as Compound Quantitative Reasoning. *** Even in Compound Quantitative Reasoning itself there are degrees of composition, and to initiate our analysis rightly we must take first the most composite type. Let us contemplate an example."

The example given is the method of reasoning pursued

by an engineer in estimating the comparative strength of bridges of different sizes. The vast amount of experience, or special knowledge, concerning the comparative strength of different materials, which the ability to solve such a problem would pre-suppose, is reduced to a minimum by taking, for example, an iron bridge, and the problems of strain are simplified by limiting the example to the tubular class of bridges. By these means the whole bearing of the example, which is made to represent, as the foregoing quotation shows, the most complex form of "Compound Quantitative Reasoning," is the joint application of two problems in mechanics to the building of bridges. The first of the propositions can be stated as follows: The bulks of similar masses of matter are to each other as the cubes of their linear dimensions, and consequently when the masses are of the same material their weights are also to each other as the cubes of their linear dimensions. This proposition, stated and explained in language familiar to all, is this: to determine the differences between masses, agree upon a unit of mass, the most convenient form of which has been found to be a cube, or a solid of equal linear dimensions. Since the length, breadth, and thickness of this unit of mass are equal, its edges or lines are equal, so that a comparison between the total number of the cubic-shaped units in each mass can be made by comparing the linear dimensions, providing the number of linearunits in the linear dimensions is first made to agree with the number of cubic units in the respective masses. The problem states that the number of linear units in the three dimensions multiplied together (or cubed in case the dimensions are equal) will equal the number of cubic units in the respective masses, or that the masses are to each other as the cubes of the linear dimensions. The stages, therefore, in this first of the two problems, the joint use of which is cited as furnishing an example of the most complex order of "Compound Quantitative Reasoning," are progressions of equations, or equalities. All mathematical progressions are steps from one equality to another, beginning always with

those simple equalities which are evident to the senses, or sensible equations. Some savages who are unable to count, form very good ideas of the comparative bulks of masses; but until they learn to count and measure they cannot understand that numbers can be made to represent bulk. It requires no mathematical mind, however, to see that they can; and the foregoing problem, stated in terms which the unmathematical reader can at once understand, would be simply this: By multiplying together the length, breadth, and thickness of a mass, we get a number which expresses the volume of the mass in any desired units. This is the extent of the question; for it goes without saying, that if numbers are made to express the exact volumes of masses, variations in volume imply variations in numbers, and comparisons of numbers are comparisons of masses.

The second problem is not so easy to reduce to its steps of equivalence, or the equations by which its conclusions are reached. It is stated as follows: In similar masses of matter which are subject to compression or tension, or, as in this case, to the transverse strain, the power of resistance varies as the squares of the (like) linear dimensions. Here we have two things made to represent each other, or equalized, or brought to an equation, which are widely different in nature, namely, the power of resistance in a mass and a superficial measurement. For if things vary with other things, they must represent them, or be equal to them, at least in the property which forms the base of the comparison. In this case, the squares of the linear dimensions of two masses are said to vary with the power of resistance of the masses. Therefore the squares of the linear dimensions must in some way be made equal to the power of resistance of the respective masses. How is this done? There is a law in mechanics, called the law of least resistance, which locates the greatest strain in a structure in a plane. This law or rule reduced to its simplest form is, that if a tube of iron of uniform size and strength be subjected to the transverse strain of (say) its own weight, the place at which it would

break, if the strain exceeded its strength, would be a transverse section of the tube, or the plane of fracture. This transverse section, or plane of fracture, is naturally two of the linear dimensions of the tube, or mass, multiplied together, and in the case of transverse strains it would be the two transverse linear dimensions which would be multiplied together to represent this transverse section in units. of squares. Here, then, the equality of nature is established between the results of the two problems. In the first a number was made to represent the bulk and also the weight of compared masses. Since every mass has three linear dimensions, if it is desired to express these masses in common multiples, or divisions of their masses, of course these divisions of mass, or units, must have three linear dimensions; and if we would compare the aggregates of units in each mass, the calculations, or process by which these aggregates are arrived at, must be compared. Now the calculations in cases of solids or masses are cubic, or three lines multiplied together, and in cases of surfaces they are squares, or two lines multiplied together. The power of resistance of a structure to a transverse strain has been simulated in the foregoing problem by a surface, and the weight of masses by solids, so that the final comparison between the results of the two problems is simply a comparison between the methods of estimating the number of superficial units in a surface and the number of solid units in a solid: one is done by multiplying together the linear units contained in two straight lines, and the other is done by multiplying together the linear units contained in three straight lines. Now if a certain operation is performed twice to accomplish a certain purpose, and the same operation is performed three times to accomplish another purpose, it is plain that the result of the operation in the latter case will be larger than that in the former, in proportion to the size of the original operation. In other words, three times a given quantity will be more than twice the same quantity, and the difference between the results will increase in exact proportion to the

size or power of the unit employed. This is equivalent to saying that the difference between three feet and two feet is greater than the difference between three inches and two inches, or simpler still, that three is greater than two. From this simple difference, the perception of which is not an intuition, because it is a sensible fact which can be demonstrated mechanically, we can build up, by retracing the steps of the above analysis, the complex problems that homogeneous masses, and therefore their weights, are to each other as the cubes of their linear dimensions, and that the power of homogeneous masses of like proportional dimensions to resist transverse strains varies as the square of the like linear dimensions. The whole comparison grows out of the fact that the operation by which the weight is estimated is performed three times, and in the case of estimating the power it is performed but twice; and this gives us the startling result that three is greater than two!

Speaking of the above problems, Mr. Spencer says: "But now, leaving out of sight the various acts by which the premises are reached and the final inference is drawn, let us consider the nature of the cognition that the ratio between the sustaining forces in the two tubes must differ from the ratio between the destroying forces; for this cognition it is which here concerns us, as exemplifying the most complex ratiocination. There is, be it observed, no direct comparison between these two ratios. How, then, are they known to be unlike? Their unlikeness is known through the intermediation of two other ratios to which they are severally equal.

"The ratio between the sustaining forces equals the ratio 12:22. The ratio between the destroying forces equals the ratio 13:23. And, as it is seen that the ratio 12:22 is unequal to the ratio 13: 23, it is by implication seen that the ratio between the sustaining forces is unequal to the ratio between the destroying forces. What is the nature of this implication? or, rather, What is the mental act by which this implication is perceived? It is manifestly not decomposable