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which a quantity having been positive becomes by gradual decrease negative, or vice versa—these words positive and negative being used to denote directions exactly the opposite of each other (i.e. having an angle of 180° between them.) We have found therefore a critical point (zero) for a gradually decreasing quantity. Is there any such point when the quantity increases? Now that there is such a point is evident from two chains of reasoning.
First it is highly probable that what happens in the inferior limit will also take place in the superior limit. In other words that as the inferior limit of increase can be passed by making the quantity gradually increase in the opposite direction; so the superior limit, that is the limit when a quantity is immeasurably great or "infinite,” can be passed by a similar change of direction and making the quantity decrease instead of increase.
Secondly we have testimony to the practical truth of this supposition in the observation of various phenomena of nature two of which are given below. The scale of increase then has two limits --one zero, and the other its opposite (i.e., infinity,) and whenever a quantity, which has been gradually decreasing, be found to be increasing in the opposite direction, it must have passed through zero, and when vice versa, through infinity.a
Upon this subject which is very difficult both from its being rather of an algebraic than a geometrical character, and also from the failure of all symbols to express gradual growth, we will only add a few remarks from Professor Price. They are in his lucid “Treatise on Infinitesimal Calculus," which was indeed the source of many ideas contained in this little work, and to which are due all the definitions founded on the method of limits, as they are either taken from it or grounded immediately upon it.
a Infinity, of course, is not the only maximum nor zero the only minimum of a curve. This merely applies to the scale of ordinary numbers.
“It is conceived," he says, “that all geometrical quantity
” whether linear, superficial, or spatial, is from its very nature capable of increase or decrease to an infinite extent. A line may be very great nay of an infinite length, or very short; space may be very small, such as, so to speak, it would require a microscope of almost infinite power to render visible, or it may
very large. Whenever such quantities vary, they do so in accordance with the law of continuity; they cannot pass from one magnitude to another without passing through all intermediate magnitudes; they grow larger and larger, or less and less. This capability of increase or decrease is involved in our idea of geometrical quantity; it is necessary to its completion, and, if it be omitted, our notions fall short of the properties of the subject-matter of the science.”
“ Having defined a plane,” he goes on to say, " to be the limiting form of a spherical surface when the radius becomes infinitely great, it follows that the extreme positive side of the plane when continued runs into the extreme negative side; that is, having traced the plane as far as we can on the positive side, we meet it again on the negative, and, although the surface appears to be discontinuous, it is not in reality so, the positive side being continued into the negative, and the apparent discontinuity arises from the defect in our power of apprehending and symbolizing such quantities. Thus then if we have any continuous curve traced on the plane, and the curve runs off to the extreme positive side of the plane, we ought not to consider it to stop or to have points of discontinuity; but we must consider the branches of it to be continued, and must look for them on the negative side of the plane.
“ It is worth remarking how exactly our ideas of a plane coincide with the definition I have given. We speak of the surface of water as a plane, and consider it to be level ; whereas it is a portion of the surface of a sphere whose radius is very large compared with the area we take; say, 4000 miles compared with a few inches.
“So again as to our conception of a straight line. A straight
line being a particular instance of a circle” (as we shall see in chapter xx.) “is a continuous line; it does not terminate at positive infinity nor at negative infinity, but the two branches of the line are connected with one another running, if we may so speak, round the circle of which the radius is infinity and joined together.” (Price's Treatise on Infinitesimal Calculus, Vol. I., pp. 261, 262. Oxford, 1852.)
We return to the motion of a single point. Suppose there is a straight line AB, and in it a fixed point C, with a line CP,
running through it, the extremity of which gradually recedes from AB. Then as P gets further off, the angle APC becomes smaller and smaller, while AP becomes larger and larger, until we can trace it no longer (1.)
We find however on the opposite side of AB that P returns with an angle gradually increasing, and the distance AP gradually decreasing (2.) Now as the angle APC from gradually decreasing has been found to increase in the opposite direction, it must have passed through zero and similarly AP must have passed through infinity. And as AP and the angle APC vary together inversely, one that is gradually increasing as the other decreases and vice versā, their critical points must have been the same. That is, AP was equal to infinity when APC was zero-a most important result.
LET us now examine the connection between the straight line and circle. We have seen that the angle between three consecutive points in a straight line is 180°, and that the angle between three consecutive points in a circle must be less than 180°, as otherwise all points in the circle would coin
1. cide with all points in the straight line. Let there be, then, three consecutive points, P,B,C, in a circle with centre AA and radius AB. Then PB and BC may
be considered as two straight lines of inappreciable length. Let one of these straight lines, BC, have a third consecutive point, Q (1.) Then there will be an angle, PBQ, and the point P will be on the same side of the straight line
CQ as the centre A is. Now let the centre A recede from B in a straight line to A; then P
2. and C will move to P and C (since the radius is increased), and CBQ will become C'BQ'. Hence, as AB increases the angle PBQ
A will decrease, and vice versa. Suppose, then, that PBQ pass through
će zero, and gradually increase upon the opposite side of the straight line QC, then the centre A will
3. change sides also, and BA must therefore have passed through a critical point when the angle QBP passed B
-A through zero (3.) And since AB increased as the angle QBP decreased,
4. this critical point must have been infinity (4.) Hence conversely when the radius AB is of an infinite length, the B three consecutive points of the circle c coincide with the three consecutive points of the straight line for the angle QBP becomes zero. And therefore all the points in this circle coincide with all the points in the straight line.
A point is a circle, the radius of which is in the Inferior Limit; so A straight line, ,
Superior Now the radii of every circle meet of course at the centre; the radii of this circle therefore meet at a point which is infinitely distant. And since no straight line can have more than one point in common with another straight line, it follows that none of the radii of this circle can have a point in common at any measurable distance. Thus we have a number of straight lines with a new relation between them, namely that they cannot have any point in common at a definite distance.