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From the north, I say, continually falleth down great abundance of water; so this north-eastern current must at the length abruptly bow toward us south on the west side of Finmark and Norway, or else strike down south-west above Greenland, or betwixt Greenland and Iceland, into the north-west strait we speak of, as of congruence it doth, if you mark the situation of that region, and by the report of Master Frobisher experience teacheth us.
The aim of this lecture is to establish a theory of congruence. You must understand at once that congruence is a controversial question. It is the theory of measurement in space and in time. The question seems simple.
Now in the first place no change could appear either as uniform or non-uniform without involving a definite determination of the congruence for time-periods. So in appealing to familiar phenomena it allows that there is some factor in nature which we can intellectually construct as a congruence theory. It does not however say anything about it except that the laws of motion are then true.
"The first geometrical, on the origin and demonstration of the laws of the figures which produce harmonious proportions; the second, architectonical, on figurate geometry and the congruence of plane and solid regular figures; the third, properly Harmonic, on the derivation of musical proportions from figures, and on the nature and distinction of things relating to song, in opposition to the old theories; the fourth, metaphysical, psychological, and astrological, on the mental essence of Harmonics, and of their kinds in the world, especially on the harmony of rays emanating on the earth from the heavenly bodies, and on their effect in nature and on the sublunary and human soul; the fifth, astronomical and metaphysical, on the very exquisite Harmonics of the celestial motions and the origin of the excentricities in harmonious proportions."
"We think it expedient," he wrote, "that when ye shall call the lords and other captains of that our land before you, as of good congruence ye must needs do; ye, after and amongst other overtures by your wisdom then to be made, shall declare unto them the great decay, ruin, and desolation of that commodious and fertile land, for lack of politic governance and good justice; which can never be brought in order unless the unbridled sensualities of insolent folk be brought under the rule of the laws.
I look on it as no slight recommendation of the theory of nature which I am expounding to you that it gives a solution of this difficulty by pointing out the factor in nature which issues in the preeminence of one congruence relation over the indefinite herd of other such relations. The reason for this result is that nature is no longer confined within space at an instant.
By 'convention' I understand Poincaré to mean that there is nothing inherent in nature itself giving any peculiar rôle to one of these congruence relations, and that the choice of one particular relation is guided by the volitions of the mind at the other end of the sense-awareness. The principle of guidance is intellectual convenience and not natural fact.
We shall find that this discovery of definite unique properties defining perpendicularity is of critical importance in the theory of congruence which is the topic for the next lecture. I regret that it has been necessary for me in this lecture to administer such a large dose of four-dimensional geometry.
The axiom also enables us to measure time in any time-system; but does not enable us to compare times in different time-systems. The second axiom of congruence concerns parallelograms on congruent bases and between the same parallels, which have also their other pairs of sides parallel.
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