■ 332.5 H Has FOURTH ANNUAL PUBLICATION Colorado College Studies. PAPERS R AD BEFORE THE COLORADO COLLEGE SCIENTIFIC SOCIETY COLORADO SPRINGS, COLORADO, THE GAZETTE PRINTING COMPANY, COLORADO SPRINGS, COLO. UNIVERSITY OF ILLINOIS LIBRARY Class " 537 .. 5 Book Volume Ja 09-20M FOURTH ANNUAL PUBLICATION. Colorado College Studies. PAPERS READ BEFORE THE COLORADO COLLEGE SCIENTIFIC SOCIETY. COLORADO SPRINGS, COLORADO. 1893. cr ^ a \ — Dee. p . 4- b \ THE GAZETTE PRINTING COMPANY, COLORADO SPRINGS, COLO. 3 3 ^ « • • »««•* « * * * 9 • • • 4 % '• * * * • t* ! « ' '■' OfFiCEtfS, 1893 President, -------- William Strieby. Vice-Presidents, ------ ( W. P. Mustard. ( L. R. Ehrich. Secretary, ------- Florian Cajori. Treasurer, Frank H. Loud. PLACE OF MEETING, Palmer Hall, Colorado College. CONTENTS. Announcement, PAGE. - 4 The Circular Locus, ... F. H. Loud. On the Eight Lines Usually Prefixed to Horat. Serin. 1. 10, W. P. Mustard. State Bank Notes, - A ~ • ' - - 4:0 W. M. Hall. 23152 ANNOUNCEMENT. The following is a complete list of the papers read at the monthly meetings of the Colorado College Scientific Society during the past year. Three of the papers are printed in full in this pamphlet, while others will be pub- lished elsewhere. October 18 , 1892 : Folk Etymology in Latin, W. P. Mustard. November 18, 1892: A Construction for the Imaginary Points and Branches of Plane Curves, - - F. H. Loud. December 16, 1892: State Bank Notes, - W. M. Hall. January 27, 1898: Friction Tests in Water-pipes and Fire- hose, ------ W. Strieby. February 24, 1893: Prayer in a Universe of Law, - - E. S. Parsons. March 24, 1893 : Acidimetry, - - - - D. J. Carnegie. On the Eight Lines Usually Prefixed to Horat. Serin. I. 10, - - W. P. Mustard. April 28, 1893: Kant’s Theory of Space and Time, - - Marion McG. Noyes. On the Multiplication of Semi-convergent Series, - Florian Cajori. May 19, 1893: The Essential Element of Religion, - - F. R. Hastings. Multipolar Dynamos, - - - - Florian Cajori. The Circular Locus, - - F. H. Loud. < THE CIRCULAR LOCUST Geometrically Constructed to Show Imaginary Values of the Varices. . : * * „ , *5 ■' ' % 3 D D 5 , ) 5 O By FRANK: H;. £jO^ID. In the paper on “A Geometrical Construction for the Imaginary Points and Branches of Plane Curves,” which I read before this society on November 18, 1892, the gen- eral method of construction therein outlined was illus- trated by the simple example of the equation x 2 +Y 2 =r 2 ; and it was shown by a diagram how the full significance of this equation, for imaginary as well as real values of the variables, may be geometrically interpreted by the aid of additional curves, accompanying the circle described around the origin. A more full and methodical treatment of the theory of this plan of construction, in its general application, has since been contributed to the Annals of Mathematics , but at the date of the present reading has not appeared in print. It is the object of the present paper to continue somewhat further the treatment of the circular locus just mentioned, as a special case of the gen- eral theory, borrowing, from the two papers above named, as much as may be necessary to render this intelligible to a reader who has seen neither of the others. The principle of construction employed is of course based upon the well-known geometric interpretation of im- aginaries, in which a+ia is represented by a line drawn from the origin to a point a units to the right of a vertical axis and « units above a horizontal axis, called the imagi- nary and the real axis respectively. The apparatus postulated in this usual representation of imaginaries — a pair of rectangular axes in a plane — coincides precisely with that of the still more familiar Cartesian method of translating algebraic equations into 6 Colorado College Studies. geometric forms, upon which is reared the edifice of ana- lytical geometry. In the customary treatment of the latter, while imaginary values of the variables appear frequently in the algebraic work’ and are recognized as of high im- portance to the theory of curves (witness the “circular points at infinity’'), they ace excluded from geometric in- terpretation. This is not necessary. Imaginaries may be admitted, with Treat” quantities, into the constructions as well as the arguments or analytical geometry, if we are content to lay aside that habit of associating each axis with a single variable, which has resulted in the names “axis of X” and “axis of Y” respectively. Let us regard these lines rather as an axis of reals and an axis of imagi- naries, and then express the usual Cartesian method of locating a point by its coordinates, by saying that the point (x, y) means the point at the extremity of the vector x+zy. When x and Y are real, this will locate the point just as Descartes does, but if either coordinate be imaginary, the point has still a definite position in the plane. Thus if x=5— 2i and y=4 + 3z we find x + 2Y=o— 2 i+4i— 3=2+2t, and the point is situated 2 units to the right of one axis, and 2 above the other, just as if its coordinates were 2, 2. Thus, when imaginaries are admitted, any point of a plane serves for the geometric representation of an infinite number of sets of coordinates, all distinct from one an- other, and we can no longer infer from the position alone of a point what are its coordinates. If we are informed by some other means what is the imaginary part of each coordinate, then this knowledge, combined with that of the position of the point, suffices to determine the real part, and hence the coordinates in full. Thus in the instance given, if we know that x + ir=2-f 2i, and that the imagi- nary part of x is —2 i, we find by subtraction that the im- aginary part of iy is 4 i, hence that the real part of Y is 4. In like manner may be found the real part of x. An equation between the variables, such as x 2 +y 2 =?' 2 , serves to restrict the number of sets of coordinates belong- ing to any one point; never depriving any point, however, The Circular Locus. 7 of its capacity of representing at least one such set. To render visible tliis effect of the equation, it is necessary to indicate, for points in different parts of the plane, by what coordinates they may, consistently with the equation, be represented; and this is effected, as already mentioned, by indicating the imaginary part of each coordinate. If a line is drawn through all these points of the plane for which the imaginary part of x has a certain constant value, say li, another through all those points for which its value is 2 1 , etc., then this system of lines wull indicate to the eye the value of the imaginary part of x for all points of the plane. Another series of lines will indicate in the same way the manner in which the imaginary part of Y varies over the plane; and when these two things are known for any point, the real parts of the coordinates become also known, and the geometric significance of the equation is fully set forth. It is true that unless a point be situated exactly on one of the lines of each series, its coordinates are only approximately indicated; but, as will be seen, it will always be possible to draw a line of each set through any assigned point of the plane, and ascertain the value of the coefficient of i which remains constant upon that line; thus the determination of the coordinates may be made precise. To these lines I have ventured to give the name of comitants, designating the two series as accountants and ?/-comitants respectively, and affixing to either name the number or symbol denoting the constant value of the co- efficient of i in the expression for the coordinate of any point upon it. Thus the accountant p is defined to mean a line (or curve) drawn through every point for which the imaginary part of the value of x has the value ip. I will also adopt in the remainder of this paper, the rule of denoting by small capitals A, B, etc., quantities which are entirely unrestricted in value, and hence are, in general, complex; while for their real parts I use italic let- ters, and Greek letters for the coefficients of the imaginary 8 Colorado College Studies. unit. Thus A =a+ia, b=& + ?’/3, etc. The Greek letter an- swering to c is taken to be y, and that answering to Y to be rj. To return to the comitants, it is plain that one of each series is of chief importance, namely, the comitant-zero. This may be compared to the equator on a map, while the comitant +1 and the comitant — 1 are on opposite sides like parallels of latitude. The ^-comitant 0 passes through all the points for which x is real, and the ^/-comitant 0 through all those for which Y is real, so that all “real points” of the locus — that is, those having both coordinates real — must lie on both these lines. For all real curves, there- fore, the comitants zero will have a common arc, and this arc will be — or, at least, will include — the curve itself, as known to the ordinary constructions of analytical geom- etry. Thus for the equation of our present example, x 2 + Y 2 =r 2 , the ^-comitant 0 consists of the circle of radius r described about the origin, and in addition of a line coin- ciding with the real axis. The ^/-comitant 0 consists of the same circle with the addition of the axis of imagina- ries.* The other comitants, of either series, are cubic curves situated on opposite sides of the axis named, and having oval branches within the circle, as will shortly be seen. A rule for constructing the comitants by points may be derived by inspection of the equation. For the £c-comi- tants, we may put the latter in the form — y 2 =x 2 — r 2 , when we have, by extraction of the square root, and the addition of x to each member, x-H'y=x±\/ ( x + r) (x — r). The mean proportional between x + r and x— r, which, as the radical indicates, is to be found, is to be understood of course not merely as having the length implied, in the Euclidean use of the word, but as bisecting the angle at * Beside having the circle in common, these two comitants zero intersect also at the origin, for this point of the locus may be regarded as having either of its co-ordinates real, but since the latter cannot both be real at once, the origin is not included in the real part of the locus. The Circular Locus. 9 the origin formed by the two vectors x + r and x— r. Simi- larly, the principle of vector-addition is to be borne in mind in uniting the result with the term x. The ordinary geometric constructions of a bisectrix and a mean propor- tional may be combined with a fair degree of convenience as follows: Let the two axes, OJ and 01, and the circle of radius r about the origin be first drawn, in ink, or so as to remain permanently through the construc- tion. If the ££-comitant p is to be drawn, where p is a given quantity, let a point M be taken on the axis of imaginaries, at a distance p above the origin; also a point S midway between O and M; and let lines MN and ST be drawn par - allel to the real axis to an indefinite length. These two lines will remain during the construction of a single comi- tant, but all that follow must be erased and drawn anew for each point constructed. On MN take any point X, and on the same line two points, H and K, at distances equal to r, on opposite sides of X. Of these, let H be on the side of X opposite M. Draw OH. With center O and radius OK describe an arc intersecting OH at G, and at G erect a perpendicular to OH. From center U (the point where OH meets ST) with radius UH, describe an arc cut- ting this perpendicular at Q. From K and G, with equal 10 Colorado College Studies. radii of any convenient length, describe arcs meeting at F; draw OF, and on it take OL and OL' in opposite direc- tions, each equal to OQ.* On the side of LOL' toward X, describe arcs from centers L and L', with radius OX; and intersect each of them by an arc drawn from center X with radius OL. Then P and P', the points of intersection, will be points on the comitant curve, one on the infinite branch, the other on the oval. Two more points on the same comi- tant are situated symmetrically to these, on the opposite side of the imaginary axis. Also, the aucomitant —;j. is equal to the ic-comitant /*, and the real axis is an axis of symmetry to the two curves. Further, if either of them be rotated through a right angle, a ?/-comitant is produced; but it must be noted that while, for all positive values of fi, the infinite branch of the a?-comitant ;j. is situated above the real axis, that of the ^/-comitant / x is to the left of the axis of imaginaries. Thus a single construction virtually determines sixteen points — four on each of two a?-comi- tants and as many more on two ?/-comitants. Each comitant, excepting the comitants zero, proves when drawm to consist of a conchoidal and an oval branch, the former approaching at infinity a line drawn parallel to an axis (the real axis in the case of aucomitants) and at a distance therefrom equal to double the coefficient (//.) of which distinguishes the particular comitant; while the oval lies on the opposite side of this axis, touching the latter at the origin. The breadth of each branch, in a direction perpendicular to the above-named axis, is the same, and is easily found to be u 2 +/P— ,u; while r 2 +/ ^ 2 +/ jt is the greatest ordinate of the curve, belonging to the point where the conchoidal branch cuts the other axis, which divides it symmetrically. Regarding together all the comitants of one series, it is apparent that all have a common point at * Another method of finding a geometric mean between two lines which ex- tend from a common point is as follows : First, bisect the angle of the lines and also its adjacent supplementary angle. On the latter bisectrix lay off half the longer line, and from its extremity, in both directions, half the shorter, thus obtaining the half sum and half difference of the lines. From the extremity of the half-difference, with radius equal to the half sum, cut off on the bisectrix of the angle a segment, which will be the required mean proportional. The Circular Locus. 11 infinity, also all the ovals are mutually tangent at the origin. Beginning with a very small value of /*, the corre- sponding comitant of course lies near the comitant zero, which consists of a circle and a line; and it is seen that the conchoidal part of the comitant answers to one semi- circumference and to that part of the line which is outside the circle, and lies just outside these parts of the comitant zero. The oval, on the other hand, answers to the other half the circumference and to its diameter, and lies inside the semicircle close against the boundary. Accordingly, its curvature is rapid on the side remote from the origin, while near the latter it is almost straight. The succeeding comitants have their conchoidal parts successively further and further from the axis, each encompassing its predeces- sor, and more nearly straight than it, while the ovals lie each one within the preceding, and are rounder as well as smaller. Hence for \j.— qc, the comitant should consist of a point (the origin), and a line at an infinite distance. We are thus able to construct and describe the systems of comitants from the equation x 2 -f Y 2 =r 2 alone; but if we wish to determine the order of these curves, so as to study them in the light of related forms, it becomes de- sirable to pass to new equations. In the article already mentioned, contributed to the Annals of Mathematics, it is shown that from the equation of any locus we may derive that of another, whose real part shall coincide with any specified comitant curve of the given locus. In the present instance we shall obtain, for the general ^-comitant — the ic-comitant £ — the following result: x 2 y+y 3 —r 2 y—2c (x 2 -\-y 2 ) = 0; and for the ?/-comitant r h x*+xy 2 —r 2 x+ 2rj (x 2 +y 2 ) = 0. These may be called the “equations of the comitants,” for the sake of brevity of expression, if it be borne in mind that from our present point of view the only equation which truly represents the comitants and nothing more , is 12 Colorado College Studies. that which at the same time represents them all, viz., the equation x 2 +Y 2 =r 2 . The new equations show that all the comitants are cubics; and the existence, as previously ascertained, of an oval branch in each, at once refers them to that one of the genera of Cayley and Salmon which consists of the projec- tions of the “ parabola cum ovali If a more minute identification is desired, a transformation of coordinates, easily effected, throws either equation into the standard form discussed by Sir Isaac Newton in his “ Enumeratio Lineavum Tertii Ordinis .” The curves thus prove to be of the kind described by him as “ defective hyperbolas having a diameter,” and to belong to the species numbered 40, whose distinguishing mark is the presence of the oval on the concave side of the conchoidal branch. To write either equation in the form adopted by Plucker as a standard, no change of axes is necessary, but the equation of the accountants, for instance, becomes by a purely algebraic modification, ( 2/ -2e)[^+^-0]-^2r#+0 2/+ |r^=O. In this expression, the existence of the asymptote ( 0-2 0) and of the “ asymptotic point ” becomes manifest, as also the position of the “satellite line” whose / ?" 2 \ 1 equation is ^2r£-f — j?/=— r 2 £: Among the diametral de- fective hyperbolas — or, to use language better correspond- ing to the vocabulary of Plucker, among cubics whose single asymptote is osculating — the group numbered xxxiv is distinguished by the fact that that line passes through the asymptotic point, while the next preceding group, xxxiii, differs from all others in which the asymptote and asymp- totic point are separate, by the fact that the satellite line does not cut the curve. In the special case in which £ has the value — , the comitant belongs to xxxiv, otherwise to 4 xxxiii. In either case, the existence and position of the The Circular Locus. 13 oval determine the species, in the classification of PI ticker V as in that of Newton; so that if c=— , the curve is of 4 species 152, but, for other values of of species 148. The equations of the comitants also suggest a process of geometrical construction entirely different from that deduced from the original equation of the locus. (See Fig. II. ) If, as before, it is desired to construct the a?-comitant fi, let the axes and the circle of radius r and center O be again drawn, and, in addition, a semicircle having for its diameter that radius, OC, of the former circle which extends along the imaginary axis in the direction indicated by the sign of //. Draw also, through B, one of the points in which the circle of radius r meets the real axis, a tangent to this circle. All the foregoing may remain unerased during the entire construction. On the tangent last drawn, take points G and H, so that BG=/j- and BH=2/^.. Draw OG, and pro- duce it sufficiently, so that a distance equal to GB may be laid off from G on the produced part, to E, while the same distance from G toward O, fixes the point D. Then OE will be the maximum distance of the conchoidal branch and OD that of the oval branch, from the axis of reals, and the extremities of these maximum ordinates are to be located on the other axis, in opposite directions from the 14 Colorado College Studies. origin. In finding other points, the construction must proceed somewhat differently for the two branches. For the infinite branch, take on the line BG a distance BK greater than BH and less than OE. Draw BC and OK, and' call their point of intersection F. Draw HF, inter- secting the imaginary axis at L, and then draw LM paral- lel to the real axis and meeting the semi-circumference, ^of radius ^ at M. Draw OM, and produce it to meet the circle of radius r at N. Draw NQ, parallel to MC, meeting the imaginary axis at Q. Finally, describe about O, with radius OQ, an arc meeting at P and P' a parallel to the real axis drawn through Iv. Then P and P' are points of the conchoidal branch. For the oval, extend the tangent GB on the opposite side of the axis of reals, and take BK' in that direction, of any length not exceeding OD. Draw OK' and CH, to in- tersect at F ' , and then F ' B, meeting the imaginary axis at S. Through S, a parallel to the real axis is to be drawn, meeting the semi-circumference at T. Then an arc, with radius OT, described about O, will meet the parallel to the real axis drawn through K',in points belonging to the oval. If a parallel ruler is used in drawing, this construction may prove more convenient than the preceding, since the compasses will have to be adjusted only once in locating each pair of new points. It should be mentioned that after the distances OD and OE have been determined, we may, if convenience require, replace the points B, G, H, K, K', on a tangent to the circle, by points at equal heights on any other parallel to the imaginary axis. In each of the foregoing constructions for a comitant fi it has been assumed that the value of t*. is directly given. An important modification of the problem may be stated as follows: Through any given point of the plane to draw the comitants of a circular locus of given radius, whose center is at the origin. Here it is necessary to de- termine at the outset the quantity (coefficient of i) which The Circular Locus. 15 characterizes each of the required comitants, and for this purpose the comitant equations may be employed. Thus, in the case of an a?-comitant, the equation may be written y (x 2 4- y 2 — v* j 2£=— — ’ a ^ ormu ^ a which directly suggests the following geometric process: Draw by the usual methods the polar of the given point with respect to the circle of radius r, and from the point in which this polar meets the line joining the given point to the origin draw a parallel to the real axis, then the perpendicular distance from this line to the given point is equal in absolute magnitude to double the quantity £, answering to the u of the preceding constructions. Having found this quantity, the comitant is constructed as before. It is to be observed that when £ and t\ are found by this process, the coordinates of the given point are fully known, hence this construction solves also the problem: To deter- mine geometrically the coordinates of a given point of a plane, when regarded as a point of a circular locus of given radius, having the origin as center. Having now sufficiently considered the form and posi- tion of the comitant curves which make up the circular locus, the next step will be to examine the most important properties of the locus as they are discussed in the elemen- tary analytical geometry, and observe what new light is shed by the present construction upon the familiar pro- cesses and results. “A right line,” it is commonly said, intersects the circle in two points, which may be real and discrete, coincident, or imaginary. Just as the “circle,” here used to mean the geometric equivalent of the equation, is in the present in- terpretation replaced by a double system of curves, together forming the circular locus , so the right line gives place to a double system of lines. These intersect the comitants of the circular locus in every part of the plane, but since these indefinitely numerous intersections of the separate comitants are not liable to be confused with the two special points mentioned in the theorem, the word “intersect” 16 Colorado College Studies. may be retained, and the proposition will read, “A circular and a linear locus intersect at two points.” By a point of intersection of two loci must of course be meant one whose coordinates, when it is regarded as a point of the one locus, are the same as when it is taken to belong to the other. If such a point, then, lies on the ^-comitant £ of one locus, it is on the a?-comitant £ of the other (the £ having the same value for the two), and a similar statement is true of the other system of comitants. Thus a point of intersec- tion of two loci may be defined as a point in which tioo comitants (viz., one of each series ) belonging to one locus are met by the corresponding comitants of the other locus. The theorem of analytical geometry, above quoted, speci- fies not only the number of intersections of the two loci, but their kinds. This, however, is done under the tacit assumption that both loci are real. The full statement intended, therefore, is as follows: “A circular and a linear locus, each of which has a real branch, intersect at two points, which may be real and discrete, real and coincident, or conjugate imaginary.” The locus of a linear equation with real coefficients (called, for brevity, a real linear locus) has the two comitants zero coinciding in a single right line, while each of the series of comitants consists of lines parallel to this. Beal intersections, whether in dis- crete or consecutive points, are formed by the real parts of the two loci, just as in the ordinary constructions of analytical geometry. The situation of the imaginary inter- sections may be easily studied by combining the two equa- tions x 2 + Y 2 =r 2 and x cos r. Elimination gives x—p cos )y— r 2 = 0 into two (since the sums of the real and imaginary terms must separately vanish), which will be mx-\-ny— r 2 =0 and [j.x-\-vy= 0. The real point of the locus is then at the intersection of two lines, whereof the former is the polar of a known point m, n; and the latter is a line through the origin, the tangent of whose inclination to the real axis is — — . With the determination of this point, two points of each of the comitants zero become known, and hence these lines may be drawn. To fully construct the locus, how- ever, it is needful to have the coordinates of one imaginary point, that the distance between successive comitants *In this is included the solution of a problem relating to a comitant curve considered as a variety of the cubic, and independently of its connection with the other comitants forming the locus ; viz : To draw a tangent at a given point of the curve. The Circular Locus. 21 of each series may be known. Following the analogy of the previous part of the problem, we may inquire for the point, both of whose coordinates are pure im- aginaries. Writing for x and ir\ for Y, the equation becomes (m+^) — r 2 =0 and is resolved into — fig—vt / m 2 + n 2 , a mean proportional between m-M'n and m— in; then >/ r 2 — M 2 — N 2 , 22 Colorado College Studies. a mean proportional between r-f ^ m 2 + n 2 and r—^ M 2 -f-N 2 , and finally a fourth proportional to M — in, r, and r±\/ r 2 — M 2 — N 2 . If P is the given point, whose coordinates are M, N, and if M is the point M, 0, then MP is equal to in, and by pro- ducing PM to Q, making MQ=PM, the point m, — n, is found. Hence the first step is accomplished when a dis- tance equal in magnitude to a mean proportional between OP and OQ is laid off both ways from O on the line bisect- ing the angle POQ. From each of the points thus fixed a distance r is then to be measured to the right, parallel to the real axis, so fixing the points H, K. Now the angle HOK is to be treated as was POQ; i. e., on its bisector is to be laid off in each direction from O a length equal to a mean proportional between its sides, and from each of the points so found a distance r measured to the right, to the points E and F respectively. If B be the right-hand ex- tremity of that diameter of the circle which lies on the real axis, we have next to make a triangle OET similar to OQE, on the base OE homologous to OQ (and with the angle TOE equal in sense as well as in magnitude to EOQ), and T will be one of the required points of contact; the other having the same relation to OF that T has to OE. (Figure omitted.) The remark has been already made that any linear locus which does not contain the point 0, 0, may be re- garded as represented by the equation MX-f-NY=r 2 ; hence the foregoing construction of the intersections of such loci needs only to be supplemented by discussing those of the locus CY=sx, to complete the treatment of linear intersec- tions. Let the points C and S be the extremities of vectors extending from O and having the values c and s respec- tively, and let P be the point whose (imaginary) coordi- nates are c, s. Produce PC to Q making CQ=PC, then the vectors OP, OQ are c-H’s and c—is respectively; so that OL and OL' (denoted by ±l) will represent =t\/c 2 -|-s 2 , if L and L ' be taken on the bisector of the angle POQ and at distances each way from O equal to the mean propor- tional. between the lengths OP, OQ. The coordinate x of The Circular Locus. 23 the point of intersection is now to be found as a third pro- portional to ±l, c and r (that is, the triangle ROX is to be made similar to LOG), and similarly Y is a third pro- portional to L, s and R. (Figure omitted.) The quantity x+it might, of course, be found at once, as in previous cases; but the construction of the separate coordinates may be more useful.* It has now been shown how to find the intersection of the circular locus by any linear locus whatever; a brief mention should be made, however, of another mode of attacking this problem, which may be best presented by considering first the simpler one, to find the point of inter- section of two given linear loci. Suppose that, in each, the two comitants zero are given in position, also in each, the £C-comitant £ (the £ having the same value in one as in the other), and in each, the ?/-comitant fj. Join the inter- section of the a?-comitants 0 to that of the accountants £. Now, since in either locus the distances from the accoun- tant 0 to any two accountants, £ and £', are as the numbers £ and £■' , it follows that the line just drawn will pass through the intersections of any two like accountants of the two loci. So also the line joining the intersection of the two ?/-comitants 0 with that of the ?/-comitants rj will pass through the intersections of all pairs of like y- comi- tants. The point in which these two lines meet is obvi- ously the required point of intersection of the given linear loci. The points of intersection of any two algebraic loci whatever may be determined by an application of the same principle; that is, the curves which pass respec- tively through the intersections of like accomitants, and through those of like ^/-comitants are always algebraic curves, whose equations result from the elimination of £ or of f] from a pair of comitant equations; and their real intersections must always correspond in position with the (real or imaginary) intersections of the two given loci. But the application of this method to determine the inter- *The discussion of the intersections made with a circular locus by imaginary radii leads directly to the theory of the trigonometric functions of an imaginary variable, but this subject is much too extensive to be here entered upon. 24 Colorado College Studies. sections of the circular and linear locus is not practically appropriate, since it invokes the aid of higher curves to treat a problem which is in fact amenable to the ruler and compasses. A similar objection must hold in other cases. The problems thus far considered embrace the con- struction of linear loci as secants, tangents, or polars, when imaginary quantities enter the problem in any way what- ever; and are hence adaptable to the treatment of any specific elementary question which involves only one circle. But there is one particular case of tangents which, on ac- count of its importance as well as of the special peculiari- ties it exhibits, requires a separate investigation. This case is that of the asymptotes. The equations of the asymptotes of the locus x 2 + y 2 =?- 2 Y . y are x + iy= 0, or — =i, and x— iy= 0, or— = —i. It is at once 5 x ’ x apparent that the equation x + iy= 0 for any (finite) values of X and Y can be satisfied only at the origin; though from the Y form— =i, it appears that infinite values of x and Y might x differ numerically by any finite amount. Hence the finite comitants of the locus all pass through the origin, while any line whatever of the plane may be taken as a comitant in- finity. Now a real “line at infinity” is characterized by opposite properties, i. e., its comitants p, wdien p is any finite quantity, lie altogether in the infinitely distant region of the plane, unless the comitant zero be regarded as parallel to an axis, when any other parallel to the same, though at a ljnite distance therefrom, becomes a finite comitant. But by assuming p infinite we may identify its comitants with any lines in the finite region, irrespective of direction. Hence the first “circular point at infinity” re- garded as the intersection of these two loci, is at any finite point; and the statement that all circles pass through this one “circular point,” is only in this sense true, that one analytical expression will serve to designate for all circles the infinite coordinates which will satisfy their equations; while, as the geometric equivalent of this analytical ex- pression is indeterminate, the point in question is in fact geometrically different for different circles. It is, actually, The Circular Locus. 25 in each circle, situated at the center. Accordingly, in ex- amining the intersection of x cos = d=r, the common r — A tangent coincides with the imaginary axis. The infinite branches of the comitants are not conchoidal, but serpen- tine, crossing their asymptote (which is still parallel to the real axis, and at a distance 2£ therefrom) at a point distant by - v from the imaginary axis. In the case of the comitant zero, this crossing is at the origin, and the serpentine curve is symmetrical to that point as a center, thus belonging to the Newtonian species numbered 88. But as f increases from the value 0, the successive comitants, no longer possessing any kind of symmetry, belong at first to the species 87, as they have no oval and no singularity. There is, however, the same gradual closing into the form of a narrow-necked purse, already observed in the case ?"=0, and the comitant p is again nodate, so belonging to the 34th species. For still greater values of c, the curve is of species 33, having an oval, which, as in all cases, shrinks to a point at the origin as ? becomes infinite, the serpentine branch in the meantime straightening toward coincidence with its asymptote. The node on the ^-comitant p is at that point of the locus for which Y=0; and it may be re- marked, as true of all forms of the locus, that the four points characterized by zero values of one or the other co- ordinate are double points of the comitants of opposite name on which they fall. Yery slight and obvious modifications are alone required to adapt the first method given in the present paper for the construction of comitants, to use with any given values of r and p. ON THE EIGHT LINES USUALLY PREFIXED TO HORAT. SERM. I. 10. 1 By WILFRED F*. MUSTARD. The eight lines usually prefixed to Horace, Satires, I. 10 are found only in some of the mss. of Keller and Holder’s third class. They are unknown to the mss. of classes I and II, and to 0 and the whole family of class III. They were apparently unknown to the Scholiasts, who would surely have considered them obscure enough to require some explanation. Mavortius did not know them. In FA' and some other mss. they appear as the be- ginning of satire 10, while in fi/mp they form a continua- tion of satire 9. On this external evidence almost all the editors have condemned the lines as an interpolation, and either marked them off by brackets or omitted them altogether. 2 They appear as part of the text in Zarotto’s Milan edition, in the first and second Aldine editions, and in the Paris edition by R. Stephanus. But even in the fifteenth cen- tury Landino rejected them, and most of the older editors followed his example. Some editors have separated them from the text but prefixed them to the satire, others have printed them separately in their commentaries, while many have omitted them altogether. Thus they do not appear in ten of the Venice editions (for the omission in the first eight Landino was responsible), in Bentley’s, Wake- field’s and some twenty others. Lambin ascribes them to some ‘ semidoctus nebulo ’ who wished to explain the open- 1 This paper offers no new theory as to the meaning, authorship or date of these obscure lines. It is merely an attempt to collect and arrange the various opinions that have been expressed with regard to them. 2 1 owe the greater part of the facts presented in this and the following para- graph to Kirchner’s edition of the first book of the Satires (Leipzig, 1854), p. 142. 32 Colorado College Studies. ing word ‘nempe.’ Jacobus Cruquius barely mentions them in his commentary as the words of a ‘simius Hora- tianus.’ Bentley omits them without mention. Others have defended the lines. Gesner restored them. Valart thought they were the work of Horace. Heindorf, followed by Bothe and others, thought that Horace had written them as an introduction to this satire but after- wards threw them aside and commenced in a different tone; or that they were an unfinished introduction to some satire discovered after his death and, with the addition of the expletive words ‘ut redeam illuc,’ prefixed to Sat. I. 10, on account of the similarity of subject. Jo. Yal. Francke proposed to insert them after verse 51 of this satire, Beisig after verse 71. Morgenstern held that Horace had written the lines, but afterwards rejected them. Schmid 3 virtually said that they were the work of Horace. Apitz 4 ascribed them to Horace, but bracketed verse 8. Urlichs 5 said that the old question is really one of sub- jective feeling as to what is worthy or unworthy of Horace. He thought the lines genuine, though he admitted their obscurity and considered the text corrupt. Doderlein found nothing seriously objectionable in the lines, and was quite certain of their genuineness. He maintained that tli& fact that they are not found in many mss. does not prove them spurious; this might be the result of chance, or even of a recension by Horace himself. W. Teuffel’s 6 verdict was similar to Morgenstern’s. The text of these obscure lines is very corrupt. The mss. of most importance for determining the original reading are FA'/S'. F, the principal representative of the large third class, is the assumed common source of the ‘ gemelli Parisini

v6ry little .support, except Porphyrio’s statement that. Floras w'riter of satires, and the fact that Titius and Fiorus were both noblemen of a literary turn, and might be called ‘ equitum doctissimi.’ That either of them could be called ‘ gram- maticorum equitum doctissimus’ is by no means apparent. ‘ Loris et funibus udis.’ The mention of dora’ and dunes’ suggests a rather savage treatment of the un- known youth referred to in this line. References to the use of dunes’ for the purpose of punishment are not very numerous. Horace, however, has ‘ Hibericis peruste funi- bus latus,’ 29 on which Orelli remarks that dunes’ made from the Spanish broom were used for flogging the ma- rines. No very satisfactory explanation of the word ‘udis’ has ever been offered. It is not clear that savage masters sometimes used a moistened lash, or that a lash so treated would cause the victim more pain. Marx 30 quotes Petro- nius, 134 B, dorum in aqua,’ as inconsistent with such ex- planations. It is unfortunate that the wisdom of the scholiasts was not brought to bear upon this word; their comments would certainly have been interesting. Pss. 3 - 6 . The changes in these three lines suggested by F. Marx have been mentioned on page 35. First he empha- sizes the importance of the word ‘ exoratus ’ in the interpreta- tion of this fragment, a word which is preserved by all the best mss. of the third class. This word, he says, may here be equivalent to dhougli vainly implored for mercy,’ like ‘exorata’ in Juvenal, 6, 415, ‘vicinos humiles rapere et con- cidere loris exorata solet.’ 31 Then reading ‘puerum’ for 4 puer,’ 32 as many earlier scholars have done, he looks about 23 Epp. I. 3,9. ™Epod. 4, 3. 30 Rhein. Mus. XLI. p. 552. 31 A similar use of ‘ exorare,’ which he might have quoted, is found in Hor. Epp. 1. 1, 6, ‘latet abditus agro, ne populum extrema toties exoret harena.’ With this meaning of * exoret,’ ‘ toties ’ may be taken literally. 32 An easy change paleographically. 40 . Colorado College Studies. for a finite verb.Q^ tsfiikmg ’ pr ‘ cutting.’ This, he thinks, is lurking in ‘udis, r which is certainly very wbak and has never been well explained. The verb is probably ‘ ussit.’ It should bp noticed that l th*e , wof*d ‘udis’ appears ‘in r as. ft? and that very bir 4 tf5n jin mss. the termination ‘-it ’ shows a medial ‘d.’ 33 For similar, uses of the verb ‘ urere ’ cp. Horace, Epp. I. 16, 47, ‘loris non ureris’; Epod. 4, 3, ‘Hibericis peruste funibus’; Sat. II. 7, 58, ‘virgis uri.’ The conjec- ture ‘quo melior versu est’ in the fourth line he puts for- ward with less confidence. Marx then refers his new reading, ‘qui multum puerum . . . ussit exoratus,’ to Vettius Philocomus, Cato’s teacher, who was one of the first to revise the work of Lucilius. 34 This man, as being ‘Lucilii familiaris,’ and possibly the same person who was censured by the poet ‘propter sermonem parum urbanum,’ 35 may have been like Aelius Stilo and Servius Clodius, a Roman knight. His name, however, suggests a Greek origin, and in the absence of any special statement as to his rank, it is not easy to assume that he was an ‘eques.’ Vs. 8. The words ‘grammaticorum equitum doctissi- mus’ are very difficult both in reference and in meaning. They would most naturally refer to the same person as ‘qui . . . exoratus,’ but they can hardly apply to the per- son who is so unfavorably compared with Cato. Schutz claims that such irony as this is quite impossible here, and failing to find any other person to whom the epithet could easily be referred, would strike out the words altogether. Apitz 36 bracketed the whole of verse 8. Kirchner and Doderlein would refer ‘doctissimus’ to the same person as ‘melior’ and ‘subtilior,’ i. e., to Cato. 33 Examples of this interchange in Horatian mss. are cited by Keller and Holder, Epilegom. III. p. 853. A similar list is given in Mayor’s The Latin Hepta- teuch , p. 251. 34 Sueton. De Gramm. 2. 35 Quint. Inst. Or. I. 5, 56, taceo de tuscis et sabinis et praenestinis quoque : nam et eorum sermone utentem Vettium (Vectium?) Lucilius insectatur, quem- admodum Pollio repreliendit in Livio Patavinitatem, licet omnia italica pro romanis habeam. 36 Coniectan. in Q. H. F. Satiras, 1856, p. 86. Horat. Serm. I. 10 (1-8). 41 The long separation is decidedly against this, and, besides, Cato could hardly be called an ‘eques.’ According to Suetonius, De Gramm., 11, his social position was doubtful in his manhood and he probably never had a knight’s in- come in his old age. To meet this last difficulty Kirchner proposed to read ‘equidem’ for ‘equitum.’ The reading ‘doctissime’ has been proposed, but this is obviously suggested by the knowledge that Lucilius was a knight, and the objectionable interval is only in- creased. The words ‘grammaticorum equitum’ are especially obscure. As they stand they would seem to imply a class of knights who were grammarians, or of grammarians who were knights, 37 but such guilds are quite unknown. Doderlein punctuated with a comma after ‘ grammati- corum.’ As has been mentioned above, he considered these eight verses the genuine introduction to Sat. I. 10, so that in trying to avoid one difficulty he created another almost as serious, by making Horace class himself among the grammarians — ‘fastidia nostra grammaticorum.’ 38 Badius Ascensis thought Maecenas was the ‘eques’; another old scholar thought of Laberius. Orelli came to the conclusion that the writer of these verses, whoever he was, knew no more who the ‘eques’ was than we do. ‘Ut redeam illuc.’ Cp. Sat. I. 1, 108, ‘illuc, unde abii, redeo,’ and Nepos, Dion., 4, ‘sed illuc revertor’; Agesil. 4, ‘sed illuc redeamus.’ It is hard to find anything in the preceding lines to which ‘illuc’ can well be referred. As Kruger 39 remarks, it cannot refer to the promised proof that Lucilius is full of faults, for this promise is not fulfilled, or to the proof of his faults on Cato’s evidence, for Horace does not return to this at all. Voss and Francke made ‘illuc’ refer in a gen- eral way to Sat. I. 4, or its subject. 37 Like Juvenal, VIII. 49, nobilis indocti, ‘ a nobleman who is an ignoramus.’ 38 This is contrary to the sentiment of Epp. I. 19, 40, ‘ non ego . . . grammati- cas ambire tribus et pulpita dignor.’ 39 Drei Satiren fuer den Schulzweck erklaert , 1850, p. 17. 42 Colorado College Studies. i It seems almost certain that these three words were in- serted on account of the abrupt opening, ‘Nempe etc.’ 40 The preceding lines were probably written with the text of Sat. I. 10 on account of the similarity of subject, and some later scribe, mistaking them for the introduction to this satire, would add the words ‘ ut redeam illuc’ to serve as a bridge to the lively opening ‘Nempe incomposito dixi etc.,’ though, as Schiitz remarks, they would serve better to connect the verses with verse 2, ‘quis tarn Lucili fautor inepte est?’ The long introduction to Sat. I. 7 (followed by ‘ad Begem redeo,’ vs. 9) may have suggested the expletive words that wore felt necessary. Keller and Holder cite as similar interpolations the four lines once prefixed to the Aeneid and the ten lines at the beginning of Hesiod’s Works and Days. It is incontestable, they add, that the satire is complete without these eight verses, and that nothing is wanting at the beginning. On the contrary, the fact that Persius, the deliberate imitator of Horace, begins one of his satires (the third) with ‘nempe’ speaks for the genuineness of the introductory ‘nempe’ here. The external evidence that these eight verses are an interpolation to Sat. I. 10 is given in the first paragraph of this paper; a careful examination of them can only re- sult in the conclusion that they are not the work of Horace at all. They have been assigned to different writers and to different periods. Kirchner ascribed them to Furius Bibaculus (circ. 700 A. U. C.), arguing from Sueton. De Gramm. 11, that Valerius Cato, if still alive when Horace wrote this satire (A. U. C. 720), must have been over seventy years old, too old to be contemplating a revision of Lucilius. This argument was soon afterwards disposed of by Schmid, 41 who proved from the same section of Suetonius that Cato could not have been more than sixty-two years old in A. U. C. 720, and 40 1 Scil. ut transitus ad Horatium sit.’ Baehrens, Fragm. Poet. Roman., 1886, p. 329. uphilol.X I. p. 54. Horat. Serm. I. 10 (1-8). 43 was probably alive several years later. 42 0. Fr. Hermann ascribed them to Fannius. Lucian Muller, in his edition of Lucilius, 1872, says they were undoubtedly composed in the time of Horace, though their authorship is uncertain. These three scholars insisted on taking ‘emendare parat’ literally. Schiitz says that the writer of the fifth verse appar- ently knew not only Epod.4, 3, ‘Hibericis peruste funibus’ and 4, 11, ‘sectus flagellis . . . praeconis ad fastidium,’ but also Epp. II. 1, 70, ‘plagosum . . . Orbilium, etc.’ This epistle is assigned by Vahlen to B. C. 14, so that these verses could not have been written by Fannius or by Furius Bibaculus. He would put the composition of the fragment as late at least as the beginning of the second century A. D. Just as Tacitus 43 says that there are men in his day who prefer Lucilius to Horace, and Quintilian 44 insists that Horace’s criticism is unfair, so the unknown writer of these lines objects to Horace’s treatment of his own model, appealing to the authority of Cato, who was of course not satisfied with the work of Lucilius as he found it, but still thought it worth revising. 45 The third verse, Schiitz maintains, is not necessarily older than Sueton. De Gramm. 2. The writer may have, known Suetonius’ account of Cato and yet made him an editor not merely a student of Cato in his younger days, either by mistake or because he knew or thought he knew better. Orelli remarks that the passage has ‘antiquum colorem,’ and assigns it to the time of Fronto. Keller would put it as late as Ausonius (circ. 350 A. D.), hinting at Tetra- dius who is addressed in Auson. Ep. 15, 9, as rivalling Lucilius. 46 F. Marx, whose beautiful emendation of these lines is often referred to in this paper, says that they are impor- tant for the history of grammar at Borne and for our 42 1 vixit ad extremam senectutem.’ 43 Dial, de Or at. 23. 44 Inst. Or. X. 1, 93. 45 It would be hard to show that Horace’s estimate of Lucilius was any lower than this. 46 * rudes Camenas qui Suessae praevenis aevoque cedis, non stilo.’ 44 Colorado College Studies. knowledge of the fate of Lucilius’ poems. The whole pass- age, he insists, suggests the philologist and reviewer, who prefers Cato’s edition of Lucilius -to his master’s earlier one. There is a vast difference between the points of view of Horace and the author of these interpolated lines: the former speaks of Lucilius himself and his works, the latter of editors and editions. If it once be assumed that the words ‘emendare parat’ do not necessarily imply that these lines were written in Cato’s lifetime, it is hard to say how late they may have been composed. Whatever their age, it is quite impossible to name their author. The fragment — and it is only a fragment, for the promise in the first verse is not fulfilled — seems to have been trans- ferred to this satire from some source rather than composed as an introduction to it, to explain and complete it. Apart from the fact that the general sentiment of the lines (so far as this can be discovered) is not in accord with that of the satire to which they are unnecessarily prefixed, it is hard to see what Horace had to do with Cato’s alleged re- vision of Lucilius or with the savage treatment of the un- fortunate youth referred to in verse 5. Keller and Holder say that the ‘Urliandsclirift’ of their third class of mss. was older than Priscian, and so also this interpolation, adding, however, that while Priscian quotes the spurious lines prefixed to the Aeneid, these eight verses are not mentioned by any of the ancient commentators. STATE BANK NOTES. By W. M. HALL. The proposal to restore the privilege of note-circulation to banks outside of the national bank system, by removing the practically prohibitory ten per cent, tax, is supported chiefly by the following doctrines: I. That the probable extinction of the national bank circulation will leave a gap in the money-supply that must be filled by notes of some kind. II. That a well-guarded system of state bank notes would give us an “elastic” circulation, i. e., one that would increase with each high tide of business, and con- tract when business slackened. III. That state bank notes would give a larger perma- nent money-circulation to parts of the country that are now scantily supplied with money. IV. That the present prohibitory tax on state bank notes violates the spirit of the Constitution if not its letter, and is a dangerous encroachment upon State powers or individual liberty or both. I. — NOTES TO FILL A VACANCY. The first of these doctrines could be summarily dis- missed, in view of the well-known habits of the interna- tional flow of gold, except so far as the shrinkage of the whole money-supply of the world would affect the scale of prices a little; a shrinkage that can be avoided by other means than bank notes. Yet the recent experience of the United States with money is not only an illustration of the international flow, but it is worth examination because it offers striking and encouraging proof that the substitu- tion of coin for national bank notes is not likely to be a painful process. 46 Colorado College Studies. From the monthly estimates, made by the Treasury Department are taken the following figures, showing the amount of each kind of money in circulation on July 1 of the year named. Money held by the banks is included, for the stock they keep does not far exceed (though it does in July somewhat exceed) a reasonable reserve, which is as much a part of the needs of ordinary business as is the reserve of five dollars or fifty below which the head of a family does not permit his cash in hand to fall. Money in the Treasury is not included, because much of it is held merely to redeem certificates that are circulating outside; and because there has been a widely varying amount there, the variations of which had an unbusinesslike origin. There would be no material difference in results, so far as the pur- pose of this paper is concerned, if the money in the Treas- ury, less the backing of certificates, were included. The figures in parentheses for the true amount of national bank notes are round numbers, estimated from the reports of the Comptroller of the Currency; this needs to be distinguished from the nominal amount, because notes of surrendered circulation, being no longer an obligation upon the banks that issued them, are really certificates payable by the United States. MONEY IN CIRCULATION. (Millions of dollars.) 1879 1882 1885 1888 1890 1892 Gold* (including Gold Certificates) . . 126 363 468 512 505 550 Silver Dollars (including Certificates) 9 87 141 256 360 384 Greenbacks and Legal Tender Certif. 302 325 331 308 335 342 Notes of 1890 98 National Bank Notes, nominally 321 352 309 245 182 167 (National Bank Notes, really) (310) (315) (270) (155) (125) (140) Subsidiary Silver 67 52 44 50 54 62 Totals 825 1,179 1,293 1,371 1,436 1,603 [Copper and nickel coins are disregarded ; so is paper fractional currency, which was reckoned about' 16 millions in 1879 and only about 7 millions since, including the amount in the Treasury. The figures of subsidiary silver for 1879 and 1882 are too large, through including trade dollars.] * The figures for gold, after 1879, are often disputed as too large ; and prob- ably with good reason. But it will be seen ( page 56) that allowance for a smaller amount does not vitiate the conclusions of this paper. State Bank Notes. 47 It may here be seen clearly how the whole volume of money has responded to the needs of increased business; the growth was rapid in the revival of business following 1879, and then fell to a much lower rate, averaging little over 30 millions annually from 1882 to 1890. But the fact which now most concerns us is that the needs of business were not provided for by the creation of bank notes, nor of any kind of notes, except the note-element in the silver dollars and the notes of 1890, and a small increase in the greenbacks outside of the Treasury. More than that, the nominal bank-note circulation was reduced in the thirteen years by 154 millions, and the true bank-note circulation by about 170 millions. If 1882 and 1892 be compared, the reduction in the ten years is 185 or 175 millions; a reduc- tion greater than the amount now outstanding. That is, we have only to do once more just what we have done since 1882, and the whole of the national bank notes will be re- placed by other money. But the matter is not quite so simple, because we shall not do just what we have done since 1882. The effect of the Act of 1890 needs to be considered, and the effect of a possible repeal of that Act. Before weighing these, it is desirable to look closely at the nature of the past additions to the money-supply, in respect to their real cost. Their cost to the country is measured approximately by the export value which the gold and silver would have had if not used for money purposes here. The cost of coinage and storage, and other such minor corrections, may be disregarded, in view of the wide allowance for error that will be used in the inquiry, and of the unequivocal result. Of course the government purchases have steadily “bulled” the silver market; how much, it is not possible to know. Against whatever such enhancement of the price of silver there has been, acting as a diminution of the cost to the country (not the government only) of the silver used for money, there is a partial offset in the increase of cost to the country of all its money-metal, through the necessity of making slightly lower average prices for exported goods in order 48 Colorado College. Studies. to send them out in place of the metal withheld and thus maintain the equilibrium of foreign trade. But to put the result of the inquiry beyond suspicion, it may be prudent to allow for the one conspicuous effect through enhance- ment of price of silver; and figures taken both with and without that allowance will be limits between which the truth lies. Evidently we must include in the cost not only the im- ported metal, but the metal produced here, so far as either has been used for money. And for this purpose we may better allow for the Treasury holdings also, because if there is an increase of metal there the country has bought it by exporting goods or by abstaining from importing them. The increase in gold used for money, either out- right or through certificates, represents one large part of our expenditure to procure new money. The other large part is represented by the gold- value for export* of the silver in the added silver dollars, plus the silver bought by notes of 1890. The supply of silver dollars in the middle of 1879, including the Treasury stock, was 41 millions; in 1882, 123 millions; in 1892, 414 millions. In the figures for 1879 and 1882 are counted several millions of silver bullion, destined soon to become dollars; the much larger amount of silver bullion in 1892, and the much wider divergence of its coinage-value and cost-price, are cause for consider- ing it separately below. The increase from 1879 to 1882 was 82 millions, which cost the government as bullion about 72 millions; the international market value was a little less, but we may neglect the error. The silver that made the increase of 291 millions of silver dollars, 1882 to 1892, was substantially all bought by the middle of 1891; it cost the government about 230 millions. Using this purchase-cost as the upper limit of what the silver really cost the country, we have yet to fix the lower limit sug- gested above. Higher price of silver, caused by govern- ment buying, affects for this purpose not only the silver thus withheld from export, but the silver still exported. * This is the export-value from time to time ; not the present export-value of the accumulated mass. State Bank Notes. 49 The change of price in silver that would in any case have been used at home does not sensibly affect the cost, being a mere readjustment of domestic exchanges. The correc- tion applies, then, to the silver bought by government, added to the net silver-export still remaining (which may be a positive or negative quantity; the latter representing an import and consequent loss by the raised price). The net export of silver from 1879 to 1892 was worth about $100,000,000; from 1882 to 1892, about $80,000,000. The government purchases from 1879 to 1890 were about a quarter of the world’s product, and since 1890 more than a third, and their effect on the price must have been con- siderable; but it seems liberal to set three-quarters of the actual price as the lower limit of the price as it might have been with no government purchases except for subsidiary coinage. On that scale, the goods received in exchange for the exported silver* of 1879-92 may, at the lower limit, have cost the country $25,000,000 less than their apparent cost; for the exported silver of 1882-92, $20,000,000 less. Any such gain diminishes the cost of our use of silver for money, and corresponding deductions are incorporated in the following table, where the cost of the silver bought by the Treasury appears separately, with the same three- quarters rule used to deduce a lower limit of true cost. COST TO THE COUNTRY, IN GOLD. (Millions of dollars.) 1879-92 1882-92 Lower limit. Upper limit. Lower limit. Upper limit. Increase of Silver Dollars, 373 millions 226 302 Increase of Silver Dollars, 291 millions 172 230 Silver bought with 1890 notes, and not coined 58 77 58 77 Deduction for enhanced value of silver exported —25 —20 Increase of Goldf 418 418 157 157 Whole cost 677 797 367 464 Average cost per year 52 61 37 46(4 * It would be a needless refinement of the question to take account of the diminished home production and export, due to lower price. f Including gold coin and bullion held by the Treasury. The official estimate is 246 miHions in 1879, 507 in 1882, 664 in 1892. 50 Colorado College Studies. That is to say, the country has given full value in goods and labor for somewhere between 677 and 797 millions of its increase of money-supply since 1879; and for between 367 and 464 millions of the increase since 1882. The money-supply itself, outside of the Treasury, in- creased by 778 millions from 1879 to 1892 (see table, p. 46); but the decrease of national bank notes caused the increase of the other elements of the currency to be still larger. Gold increased 424 millions, silver dollars 375 millions, “greenbacks” 40 millions, and 98 millions of 1890 notes were added; making an increase, aside from national bank notes and pieces less than one dollar (which last would have shown about the same behavior under any system of major currency), of 937 millions. This last is the amount of money that has been added in thirteen years past to meet the needs of increased business and take the place of the declining bank note circulation. But we have seen that the country earned meanwhile, i. e., bought with goods and labor for which it received nothing else in exchange, between 677 and 797 millions.* That is, the note-elementf in the addition of 937 millions was between 140 and 260 millions. If 1882 and 1892 be compared, the whole addition of money other than national bank notes and small pieces will appear as 599 millions, and the note- * This comparison of increase of money-supply outside the Treasury with increase of money-metal both within and without the Treasury may seem irra- tional. But the former is the true measure of past additions to the money-supply and the better basis for judging what future additions are probable, and hence what the strain of making them will be ; while the sacrifice in former acquisi- tions is better measured by the addition of metal in Treasury and outside circu- lation together. If any one nevertheless prefers to compare outside circulation in both cases, he will find the increase of silver dollars (table, p. 46) from 1879 to 1892 to be 375 millions, 1890 notes 98 millions, gold 424 millions, and the resulting “ lower and upper limits ” about 700 and 825 millions ; differing from the 677 and 797 millions, reckoned above, in the direction of decreasing the note-element in past acquisitions, and therefore of decreasing the sacrifice needed in future acqui- sitions that may contain a less note-element or none at all. That is, it would strengthen the conclusion that in the text above is based upon a less favorable supposition. fNot the note-element reckoned upon the present bullion value of silver, but the unearned part of the issues of silver dollars and 1890 notes as they were made. State Bank Notes. 51 element as between 135 and 232 millions. The following figures show, accordingly, the average annual addition: Whole addition. Note-element. ( Millions.) Annually, 1879 to 1892 72 Between 11 and 20 Annually, 1882 to 1892 60 Between 1314 and 23 And the country has earned (see, also, the table on p. 49) between 52 and 61 millions annually through the longer period, and between 37 and 46^ millions annually through the shorter. It may safely be said that our probable dealing with silver in the next few years (omitting free coinage as too improbable in the immediate future to justify the discus- sion, necessarily long, of its bearing on the present ques- tion) will lie within a range bounded by — (1) Continuance of the Act of 1890. (2) Revival of the Act of 1878. (3) Purchase of silver, and issue of notes whose silver backing, reckoned as bullion, is kept equal to the face value of the notes; kept equal by subsequent purchase of silver, if necessary, without issue of notes against the supplementary silver. (4) Suspension of silver purchases, except for small coins. It is quite possible that silver legislation may combine two of these methods, or change the amount of silver to be bought under (1) or (2). But the present object is to discover whether the national sacrifice in obtaining addi- tional money will be greater hereafter than it has been for a few years past, and that object will be sufficiently attained by taking each method separately and noting the effect in (1), (2) and (3) of different amounts of silver- purchase; for any combination will be more favorable than the least favorable method standing alone. ( 1. ) Continuance of the Act of 1890. If this happens, the addition of money will be wholly earned, except for the “lower limit” purpose a note-element due to the higher price of the silver bought and the silver exported. The 52 Colorado College Studies. silver bought is 54 million ounces annually, which is sub- stantially the whole amount available for Treasury pur- chase or export.* The domestic production of silver is increasing by about 4 million ounces per year; but sup- posing (to keep on the less favorable side of probabilityf ) that the amount exported in the next dozen years should average only 6 million ounces, while 54 millions were still bought, the amount through which the higher price could operate to diminish the true cost would be about 60 million ounces annually ; worth $60,000,000 at one dollar, $45,000,000 at 75 cents. Accordingly the note-element, on the three-quarters scale, lies between zero and 15 millions in the improbable event of a rise of silver that carried it to average $1 an ounce, between zero and 11 millions if silver averaged 75 cents. A smaller government purchase would not change the quantity of silver affected, but it would of course bring the note-element nearer to the zero limit through affecting the price less. ( 2. ) Revival of the Act of 1878. Taking its minimum purchase of $24,000,000 worth of silver annually, the dol- lars coined would be, with silver at $1 an ounce, 31 mil- lions; with silver at 75 cents, 41 millions. The seigniorage would thus be 7 and 17 millions at those prices respectively. The other part of the note-element, by the three-quarters rule, would be (as under the Act of 1890) between zero and 15 millions at the former price, between zero and 11 mil- lions at the latter. The whole note-element is thus between 7 and 22 millions when silver is at one dollar an ounce, between 17 and 28 millions at 75 cents. Evidently the note-element is enlarged by increased purchases or by a fall of silver. (3.) Issue of notes with a constantly equivalent silver hacking ; a backing kept equivalent, when the price of silver declines, by purchase of more silver without issue of notes * The net import was about 3 millions in the fiscal year 1891 ; net export 6 millions in 1892. t A larger supposed export would increase the note-element and decrease the sacrifice. State Bank Notes. 53 against it. The question, highly important for other pur- poses, whether the notes are redeemable in gold or in a gold dollar’s worth of silver, has no bearing on this discussion. In either case the note-element is between zero and a quarter of the export-price of 60 million ounces, minus the cost of silver bought in case of falling price to keep up the backing of notes issued earlier — that kind of purchase being cost without addition to the money-supply. It would be an extreme supposition that silver should fall to 60 cents an ounce in the next ten years; that would be about 2^ cents annually. Such a fall would require, if the annual pur- chase for note-issue were $24,000,000, a purchase of sup- plementary silver amounting to less than a million dollars in the first year, to 10 millions in the last year, but averag- ing about 5 millions annually through the ten years; while the note-element due to upholding the price of silver would average (with silver at an average of 72 cents) between zero and 11 millions. Deducting the 5 millions of cost for supplements, we have minus 5 and plus 6 millions as the limits of the note-element; that is, it might possibly be a more expensive way of obtaining new money than import- ing gold would be.* On the less extreme supposition of a fall of silver to 70 cents in ten years, the average annual supplementary purchase (the principal purchase being still 24 millions) would be something less than 3 millions, mak- ing the note-element somewhere between minus 3 and plus 9 millions.f The increased expense, in the later years, of maintaining such a note-system in case of a progressive fall of silver is of course a serious objection to the system, unless it is believed that silver will not continue to fall. If silver does not change in price, the note-element is the same as under the Act of 1890; with silver at 84 cents, it is between zero and 12^ millions. Greater purchases would *A loss of this kind, payable in future, has already been incurred by the country through its large purchase of silver, if silver does not rise again, and if the notes of 1890 or silver dollars are ever given a 100 per cent, backing or are withdrawn and the silver sold. f Silver then averages about 77 cents, and the note-element, aside from cost of supplements, is between zero and ll l / 2 millions. 54 Colorado College Studies. probably increase the note-element unless silver declined 2 cents or more yearly. (4.) Suspension of silver purchases, except for small coins . This would leave the natural movement of gold to make the necessary increase of money, and would provide no note-element. Tabulating the effects of these methods of dealing with silver, we have the note-element appearing as follows: NOTE-ELEMENT LIES BETWEEN— EFFECT UPON THE NOTE-ELEMENT— Method (l),withActof ’90 unchanged; silver at 100 Method (1) ,with Act of ’90 unchanged; silver at 75 Method (2), with annual purchase $24,000,000 ; silver at 100 Method (2), with annual purchase $24,000,000 ; silver at 75 Method (3), with annual purchase (for note issue) $24,000,000 ; silver declining 2*4 cents yearly Method (3), with purchase 24 mill. ; silver declining 1*4 cents Method (3), with purchase 24 mill. ; no decline of silver Method (4) 0 and 15 mill. 1 of increased purchases. of cheaper silver. 0 and 11 mill. ) Increase. Decrease. 7 and 22 mill. ) Increase. Increase. 17 and 28 mill. ) -5 and +6 mill.* Uncertain. Decrease. -3 and +9 mill.* Uncertain. Decrease. 0 and 12*4 mill. Increase. Decrease. No note-element. The past annual increase of the whole money-supply, together with the money that replaced bank notes, has in- cluded a note-element lying somewhere between 11 and 23 millions (see page 51). If we continue to extinguish bank notes at the same rate, make no change in the amount of greenbacks, and increase the whole money-supply at the same rate as before, it appears from the table above that Method (2) would involve no appreciable decrease, per- haps an increase, in the note-element; that is, the money to serve the growing needs of trade and to take the place of disappearing notes would cost us not appreciably more, perhaps less, than it has in the recent past. Under Method (1) the national expense on this score would be say 8 to 12 millions more, annually, than it has been in the recent past. * For ten years only, and as an average ; the note-element being less than zero in the later years, and going further below after the ten years, if the fall of silver continued at anything like the same rate. State Bank Notes. 55 Under Method (4) it would be between 10 and 20 millions more. Under Method (3) it might in a very unfavorable case be 20 millions more, but with a slower decline of silver the added expense would be nearer the 12 millions or so (between 10 and 15) wdiich would accompany a stationary price of silver. Under Method (3 ) this average cost through ten years represents a smaller cost in the earlier years and a heavier one in the later years, if silver falls. It is only under Method (3) that we have to anticipate a cost, for the average annual addition of money and extinction of bank notes, exceeding by more than about 15 millions the cost that has already become habitual. Method (3), if silver should fall rapidly, would be burdensome after a few years, particularly if the annual issue of notes were much more than the 24 millions reckoned in the table; but no one wishes to see that method adopted if silver is to fall rapidly; and in any case, the extinction of bank notes within a dozen years would contribute only 12 or 14 mil- lions annually to the burden. Under any of the more prob- able forms of dealing with silver the sacrifice of the country, in extinguishing the bank notes while it increased the whole circulation, as usual, would not exceed by more than about 15 millions the sacrifice that is already customary; and it might not exceed that at all. Remembering that in place of an annual cost of between 40 and 60 millions for the near future, which has been the implied basis of the present reckoning because it was the average cost for the past few years (see page 49), we might have for a part of the time, as in 1879-82, a cost of 100 millions attended by great commercial prosperity — remembering, too, that it takes more than an occasional waste of 20 or 30 millions by Congress to make an appreciable difference in the course of business — it seems unlikely that the withdrawal of all the national bank notes within ten years can give a sensi- ble check to business. Indeed, the greatest expense for new money comes just at the time when the country can best afford it, in times of rapid growth of business; and just at the time when there is need of a check upon excessive speculation. 56 Colorado College Studies. The conclusion just reached has so wide a margin of safety that it excuses the omission, for simplicity’s sake, of a number of corrections; some offset each other, some are mere differences of degree of an element common to all the years (e.g. a deduction from the probably excessive Treasury estimate of gold coin in private hands), and the aggregate of the corrections can scarcely swell the difference between past and future sacrifice in the enlargement of the money- supply so as to call for the retention of national bank notes, or for the provision of any other notes to take their place. If there is any need for state bank notes, it is to be found elsewhere. II.— ELASTICITY. In popular discussion of the repeal of the bank-note tax it is often assumed, as something near an axiom, that a bank note system may readily be made “ elastic,” and it seems to be implied that the easier it is for banks to issue notes the more elastic the resulting currency will be. But among economists this is so far from being a generally accepted truth that some reputable writers deny the pos- sibility of bank notes following the needs of trade, either in expansion or contraction ( except in the same way that coin would have done), so long as the bank notes are really convertible, i. e. are promptly and willingly redeemed by the bank. If nevertheless we grant that an expansion is possible, it is reckless to assume, without careful examina- tion, that the bank-note currency would contract again when trade slackened. And unless it does so contract, what we have is not elasticity but a wholly inelastic distensibility. Entering first a protest against another too easy assump- tion, that elasticity is an unmixed good (for much may be said for the doctrine that the evil in it exceeds the good, through removing one of the barriers against speculative excitement), we have to inquire what are the causes that may limit a note-circulation, and whether any action of the State governments upon those causes can give an increase State Bank Notes. 57 or an elasticity that the National government cannot give by similar action. A bank issues notes in order to increase its receipts of interest. In a country where notes are familiar, it can usually carry notes with a reserve smaller than that which it needs to carry deposits*; and accordingly, unless taxes or other expenses intervene, its loans that can be made through notes are more profitable than its loans through book-credits of deposit. But in the United States most borrowers prefer book- credits, and the practicable note- issue is restricted to so much as can be paid out, whether in loans or in other money-payments (e.p.upon checks pre- sented), without coming back for redemption faster than it is reissued. Within that limit (beyond it, if bank notes were not convertible) the bank has a motive for keeping its circulation as large as possible, unless other expenses or hindrances appear. In the United States the limit so set would be narrow for the individual bank, on account of the large proportion of credit-transactions, except for the habitually long life of the circulating notes; in fact it is so wide that a limit for the whole note-circulation, drawn on the same scale, would be impossibly large. The real limit for the whole note-circulation is of course much lower, because as the whole amount approached a point consid- ered to be dangerous, redemptions would become more fre- quent, i. e. the limit for individual banks would shrink. Banks do not grow weary of making a profit, nor stop issu- ing notes without a reason for stopping. Where the note- circulation falls short of the limit (usually wide, when banks are well managed) set by the return of notes for re- demption, it is certain that there are definite causes for it, either prohibiting the increase of circulation or offsetting the profit by some expense, inconvenience or dread of in- jury. Including redemption, we may name the following restraints upon the issue of notes, some of which are * Strictly speaking, deposits which are caused directly or indirectly by the bank’s loans. Other deposits have no bearing on the question whether loans by book-credit or by notes are more profitable. 58 Colorado College Studies. found at work in every bank-note system, no matter how bad: checks upon the issue of bank notes. 1. Return of notes for redemption. The effect of this important restraint has been outlined above. It is thor- oughly effectual only when redemption is enforced by law and not discouraged by public opinion or by bank pressure ( e . g. refusing discounts and other bank services to persons w 7 ho have presented notes for redemption ) . But as we shall see hereafter, unless certain artificial means are used to compel the return of the notes within a given time, the action of this check does not prevent an increase from year to year till the issue is very large, if public confidence in the soundness of the banks prevails. 2. Fear of discredit. That is, a belief of the bank managers that further issue of its notes would impair the bank’s credit, and either directly fail of its purpose by bringing back for redemption notes equal to the new issue, or injure the deposits and other business of the bank. 3. Fear of general injury to business. This may in- clude one or more of the following elements: Fear of the effect on the bank’s own business of a general derangement of business caused by bad currency; fear of the effect of such derangement on the interests of the managers outside the bank; and a sense of responsibility and trusteeship towards the business community. This check is seen at its best where a single large bank issues most of the bank notes of a country (the circulation of notes of other banks being prevented or kept within narrow limits bylaw); the Bank of England, for instance, would doubtless be tem- perate in the issue of notes, even if it were not hindered by law from making any profit by an increase of circula- tion. The Bank of France is influenced by a similar com- bined responsibility and prudence. But when there are many banks, each knows that its own note-issue will be only a small part of the whole note-circulation, and that its own abstention from new issues will have little effect State Bank Notes. 59 towards reducing the whole; perhaps no effect at all, be- cause other banks will issue more instead before the more rapid return for redemption comes into play, or the legal maximum, if any, is reached. And if some banks are still conservative when the motive to be so is thus diminished, other banks will surely, in such a country as the United States, be more tempted by the profit than deterred by the possible future evil, which is made scarcely more probable by their venture; unless, indeed, the number and character of the banks be so limited by law that a joint agreement is practicable (a case which will be mentioned below). The point is, that we cannot look to this check, unless supple- mented by an agreement among the banks, to prevent an im- mediate increase of bank circulation if the latter were made otherwise profitable; for some banks would surely grasp at the profit. 4. I^egal limitation of the number of note-issuing banks, either by requiring a special legislative charter for each, or by imposing onerous conditions for going into business under general law. This check acts upon a part of the field only, of course; leaving the banks which come within the privilege free to increase circulation, except so far as other checks interfere. 5. A legal maximum of bank note circulation. This, if set low enough, is perfect insurance against an undesir- able enlargement. Evidently, however, it allows no “elas- ticity ” beyond the maximum, and does nothing to prevent the circulation rising to near the maximum, so that further elasticity is impossible unless that begins by decrease; and the decrease must then be due to other checks, and not to this, unless the maximum itself is made different for different times of year. 6. A legal minimum of reserve or of securities. So far as this minimum is greater than the amount of reserve or of securities which the banks would hold of their own accord, it is a check upon circulation by reducing the profit. In our national banks it takes the form of an amount of 60 Colorado College Studies. bonds costing much more than the reserve that the banks would voluntarily hold, with only partial compensation for the difference in the low interest received on the bonds. 7. Taxes on circulation; another mode of reducing the profit. Under the present national bank law, not only does the tax on the notes of outside banks extinguish the profit and prevent their issue, but the tax upon national banks of o neper cent, on circulation contributes materially to the well-known unprofitableness to many of them of their notes. 8. Legal requirements hampering the ready circula- tion of notes. For instance, that they shall be of large denominations only, that they shall bear interest (this cuts into the profit also), or that they shall be of cumbrous size. Requirements that they should be indorsed on pass- ing, that they should be on some easily damaged kind of paper, that they should be presented for redemption within a certain time or lose part of their face value, would have similar effect. 9. Agreement among hanks to set limits similar to 5 , 6 and 8. In view of the number of American banks and the difficulty of making and enforcing such an agreement, it would be a waste of time to discuss this as a possible check upon future issues here; at least till it is seriously proposed to have a few large banks monopolize the bank-note issue. Depreciation may in times of inconvertibility act as a check upon issue, but no one now proposes to have a sys- tem subject to enough depreciation to come within the range of that check. Another check, now practically out of the field, probably acted upon the bank-note circulation in America early in the century: scarcity of capital and high rewards for its use in other ways than banking. Some men who might have been attracted by the high returns of bank-note issue were still more attracted by other enter- prises that would employ their time so fully as to make inconvenient the attempt to conduct a bank. Turning back now to see what are the checks that keep our national bank note circulation comparatively small, State Bank Notes. 61 we observe at once that the checks numbered 2, 8, 5 and 9 have no effect upon it. There is no fear by any bank that its credit would suffer by issuing notes up to the amount permitted by law in view of its capital, no fear of general injury to business by the issue of another hundred millions or two*, no legal maximum of circulation, no agreement among the banks. Under No. 8 the omission to make the notes a general legal tender does not in practice restrict their circulation, nor would the banks issue any more notes if they were still permitted to use denominations less than five dollars; the exclusion of bank notes from the legal reserve of national banks acts towards encouraging the return of the notes for redemption, thus setting at work check No. 1. Contrariwise, the law has by its assimilation of bank notes to government notes, in size and general aspect, done a little towards promoting their ready circula- tion. No. 4, legal limitation of the number of note-issuing banks, appears in the form of “onerous conditions,” such as a minimum of capital, government inspection, prohibi- tion of real-estate loans and of all loans beyond a certain multiple of the reserve held; but this is more than off- set, on the whole, by the advantage which national banks receive in public opinion of their probable soundness, and it would not prevent an increase of note-circulation if such circulation offered a profit. We have left, therefore, the checks numbered 1, 6 and 7 ; deducting No. 1, which is not a limit but a drag upon increase (making it slower without stopping it), the true causes of the smallness of the circulation are in Nos. 6 and 7, i. e., (a) the require- ment of deposit of bonds whose market value is much higher than the amount of notes issued against them, bonds which bear a lower interest than the bank’s ordi- nary business yields and most of which depreciate (through the effect of the approach of maturity upon their premium) * Provided it were gradual, and the annual purchases of silver by the Treasury were small. If brought face to face with the possibility of a large increase of bank notes not so guarded, banks would doubtless recognize the danger; but, as suggested under No. 3 above, only a part of the banks would probably hold back from issue. 62 Colorado College Studies. before the bank can sell them, (6) the small deposit of cash for redemption purposes at Washington, (c) the one per cent, tax on the average note-circulation of the bank * From many banks these take more than the whole profit of the notes, and the banks tolerate the loss as a sort of advertising expense, in order to keep the advantage of being known to be under government inspection. The decline of the note-circulation before 1890 shows that on the average the loss exceeded all the gains, including the advertisement, and the slight increase f since 1890 shows that for two years past the gains have by a very narrow margin exceeded the loss, taking all banks together. The failure of the note-circulation to be elastic is due to the expense and trouble of buying and depositing bonds to secure the temporary addition, and withdrawing and sell- ing the bonds when the time for contraction comes. Unless the bonds are withdrawn, the bank has no motive to with- draw the added notes. Since our note-circulation is now kept small by the un- profitableness to the banks of further increase (except the trifling present increase, which may at any time fade out by disappearance of the narrow margin of gain which causes it) the way to make room for either elasticity or permanent increase is to remove part at least of the causes of the unprofitableness; to relax the checks called 6 and 7 above. Public attention seems to be given to the proposal to abolish the ten per cent, national tax on state bank cir- culation; but room for increase may be had just as cer- tainly, though not for so great an increase, by reducing or removing the one per cent, tax on national bank circulation, or reducing the required deposit of bonds, or permitting the substitution of other securities bearing higher interest. * Minor expenses (like examiner’s fees), so far as they are independent of the amount of circulation, may better be included under No. 4 ; and some which vary with the amount, such as the cost of redemption and reissue, are small enough to be neglected in this article. t Increase in the true circulation. Notes of abandoned circulation, awaiting redemption by the Treasury, have no place in such a comparison as this ; though the government reports of circulating money include them in the figures of bank notes, which thus deceptively show a decrease since 1890. State Bank Notes. 63 It would take little change of these restraints to cause a marked^ acceleration of the increase of national banknotes that is even now visible. National banks are as ready as any others to increase their circulation when the increase pays. IMPERFECT CONTRACTION. Now let us postpone all the difficulties of securing a sound state bank currency, and suppose that either by adopting a system of lightly taxed state bank notes or by loosening the restrictions upon national banks we make possible the issue of 100 millions more of secure bank notes. There is no way of making the issue possible but by making it profitable. If the checks were so lessened as to let the profit exceed them before the harvest-season* came, there is good reason to believe that a part of the extra notes thus made possible would not only be sent out but kept out without waiting for the crop-season, unless certain legal requirements, hitherto unusual, were imposed upon the banks or certain new conditions attached to the notes. If this occurred long enough before the crop-season to permit the swelling of the currency to affect the amount of coin (either by increasing export or diminishing import of money-metal), it would to that extent merely substitute notes for coin lost; the remainder of the “slack” would be taken up by the growth of business at the crop-season, and this remainder only would have any value towards giving elasticity, the former part being a needless and ( if extensive or often repeated) an injurious enlargement of the bank- note circulation. But we will take a case more favorable to the proposal of a bank note elasticity, and suppose no such wasted ele- ment in the new issue; suppose the checks to be so skill- fully balanced against the profit that the latter does not emerge superior till the rate of short-time interest rises at harvest-time. The new notes then go out, and the first half of elasticity is thus displayed. But what assurance *For simplicity, the autumnal moving of the crops is spoken of alone, here and later in the discussion ; but the same arguments apply to the spring trade and to sporadic maxima of business. 64 Colorado College Studies. have we that when the crop-season is past the excess of notes will be redeemed and disappear ? The apparent answer is that with slackening of business the currency will be redundant, bank reserves will be overstocked, and the banks will have no profitable use for the excess of notes, so that redemptions will exceed reissues till the whole redundancy is withdrawn. But this is only an apparent answer. The redundancy after the crop season is not identical with the preceding increase of notes; it may be far less; it usually would be far less in the United States. The growth of use for money at crop-moving time is in this country a mere wave in an ascending slope, a slope ascending so rapidly that we seldom add less than 20 millions a year to our stock of money, and sometimes 100 millions; the use is never, except through unusual business depression, so low after the wave as before it. Almost always, there- fore, if we provided elasticity by means of bank notes unattended by special appliances for forcing their sub- sequent withdrawal, and if the elasticity were wholly by bank notes and no increase of coin shared in it, the re- dundancy of money after the crop-season would be less than the increase of notes during the crop-season. Con- ceding that notes equivalent to the whole redundancy were redeemed and retired, what motive would the banks have for retiring the rest of the recently added notes? Reserves have sunk to their normal size, for that is what the disap- pearance of redundancy means; the banks have no more money than they want to use. As fast as notes are re- deemed, it is then the bank’s interest to reissue them, i. e. to use the notes in place of the money paid out in redeem- ing them. There is thus a part of the recently added note-supply that will remain in circulation. This process would be repeated at each maximum of business, and ac- cordingly, without such special devices as will presently be mentioned, an elasticity provided by bank notes alone would cause an increase of the bank note currency from year to year. But it may be that a part of the increase of money at crop-moving time is in coin. By taking the effect of the State Bank Notes. 65 mixture upon a period of several years as a whole, we may eliminate the causes of complication and find a practical certainty that the note-circulation would show a progres- sive increase. For all the effects upon the coin-supply that the occasional presence of more notes can have are upon one side, in the direction of making the coin-supply less than it would otherwise have been. The issue of more notes at the crop-season will check the rise of short-time interest and make money easier; in fact, that is what it is intended for. Easier money means a better maintenance of prices of goods, and consequently less encouragement of purchases here by foreign buyers and less tendency to start an import of gold or a diminution of its export. In short, the pressure at the crop-season, so far as it is now relieved by any retention or import of gold, would force less relief of that kind because the pressure itself would be less. This, in turn, is only a part of the more general statement that whenever there is any change in our money- stock of gold, the presence of recently added bank notes in the circulation tends to make the increase of gold less or the decrease more than it would otherwise have been; for if any given quantity of these bank notes had been ab- sent, the tightness of the money-market would have been increased or its plethora diminished. Taking any period of years, we may rely upon it that effects of this kind will have happened while the extra bank notes of the crop- season were afloat, in which case bank notes will have taken the place of the coin expelled. And these changes will be cumulative, because the presence of bank notes always acts on that side when it acts at all. MEANS OF ENFORCING CONTRACTION. The importance of this practical certainty that the bank note supply will increase from year to year if the checks are so balanced against profit as to permit an easy increase of notes at every maximum of business, and if no special devices are used to force all the new notes back when busi- ness slackens, lies in its strongly commending to us the use of such special devices if we undertake to make an 66 Colorado College Studies. elastic note-circulation. That a progressive increase of notes is, on the whole, an evil, most economists will agree. It is true that there is a gain by use of cheap money when it is good money; that is, by the less sacrifice of goods involved in obtaining it, the relative diminution of the earned part and increase of the note-part. But that seems to be more than offset by the risks that attend a steady increase of bank note money — sometimes a distant risk of an amount great enough to be depreciated, always a risk of corruption of popular judgment about money through suggestion of government paper as a resource at the first pinch, and usually an enhancement (through enlarging the number of issuing banks and spreading the habit of extra note-issues) of the greater risk of encourag- ing riotous speculation that attends all schemes for elas- ticity by bank notes. As a special device to keep an elastic note-circulation from increasing progressively, the ingenious suggestion of the late John Jay Knox * may be considered first: that the notes added at times of increase should be of a different color from the others, and should bear interest after a cer- tain date. This proposal takes advantage of the well known fact that notes bearing interest will not circulate as money except when money is very scarce; the holders naturally treat them as a sort of bond instead. Mr. Knox’s species of notes would, accordingly, almost all drop out of circula- tion as money soon after the date when interest was to begin. If the notes bore a low rate of interest, they would find their way to the issuing bank before long; if a high rate, part of them would be kept by the holders as an in- vestment. In the case of a high rate, indeed, the disappear- ance from circulation would begin before the date arrived, and perhaps so soon as to impair the intended elasticity of the note-circulation. While trusting to the behavior of the note-holders is reasonably safe, it seems simpler and more certainly effect- ual in every case to provide a legal limit acting directly * See the Forum , February, 1892. State Bank Notes. 67 upon the banks. Suppose, for instance, that the increase of the ordinary note-circulation is prevented by fixing a maximum for it as nearly as possible at the point where it now is, admitting hereafter no new issue of notes on the ordinary terms, except in place of such circulation with- drawn;* and that extra circulation be permitted on some such terms as these — the extra notes to be issued only dur- ing two or three specified months just before and during the crop-season, and the whole circulation of the bank to be reduced to its former dimensions by, say, January 15 following. The reduction, or its equivalent, may be made certain by requiring from the bank a deposit of money on January 15 in place of any excess of notes still outstand- ing; the government either to undertake the subsequent redemption itself, or to return the money to the bank as fast as the redemption proceeded. The money being kept idle by the government meanwhile, the effect on the money- supply of the country is the same as if the excess of notes had been wholly retired before January 15; and the con- traction has begun some time before the date, as the bank, besides its actual redemption, began to accumulate money for the final alternative redemption or deposit with the government. The extra notes could then have precisely the same appearance as the ordinary notes, and have the same validity as between the bank and the holder; it would make no difference whether the notes redeemed and re- tired were the same individual notes that went out in the extra issue; all that is necessary is the retirement of a certain amount of the notes of that bank, drawn indiffer- ently from the old and the new ones. The whole matter would be settled between the government and the bank, and no one else would be troubled with any discrimination among the notes he handles. A second period of possible issue may be provided for the “spring trade”; but there should be at least one space of a month or two in each * Giving preference to newly organized banks in allotting this substituted ordinary circulation; if any were left, old banks that wished to enlarge their ordinary circulation could have it. 68 Colorado College Studies. year, and better two such spaces, when the extra circula- tion is wholly absent, so that it may have no opportunity to become permanent circulation and displace coin.* The hard-and-fast limiting of the ordinary note-circulation need not exclude new banks from the privilege of note- issue, for if there are not enough old banks withdrawing ordinary circulation to make room for them in the ordinary circulation, the new banks may nevertheless, by coming under the inspection-rules, be allowed to share in the extra issues. The question how the extra notes should be secured is of some importance. If deposit of securities is required, it will be found to impair the readiness of the banks to issue extra notes; the nearer the formality and expense of making the issue approached that of the ordinary issue, the less elasticity the system would give. On the other hand, if the original security were no more than enough for the ordinary issue, it would not be enough for a per- fect safeguard to the increased amount.f A first claim upon all the property of the bank, reinforced by a safety fund and perhaps by a personal liability of the stock- holders, would be very strong, but might imaginably fail in some one or two instances at last; though the govern- ment might protect the note-holder by assuming that risk itself through guarantying the notes, or transfer it to the other banks in the system through making them respon- sible beyond their share of the safety fund. All these comments upon the mode of securing the notes' apply equally to Mr. Knox’s plan and to the second plan de- scribed. Bo do all the following questions: What securi- ties should take the place of United States bonds when those are extinct; whether a maximum should be set to the * I. e. displace coin progressively. There would doubtless be a slight dis- placement in the first year or two, and then the coin-supply would go on a little less than it would otherwise have been, but less by a nearly constant amount. t But the deposit of securities for the ordinary circulation might be made to exceed the face-value of the ordinary notes by a margin large enough to cover a considerable extra circulation. Just now, national banks whose deposited bonds are 4 per cents could issue 30 per cent, more circulation without exceeding the market value of their bonds plus their redemption-deposit of cash. State Bank Notes. 09 extra notes also; whether all banks in the system should have the extra-note privilege, or banks in certain cities only; what features of the law should be left subject to variation by the Comptroller of the Currency or some other executive officer. There ought to be an arbitrary maxi- mum of increase, lest in an occasional time of speculative temptation the note-issue should run wild; for serious harm might be done by a rapid increase long before it reached any automatic check by redemption. Both of the plans need free criticism and working out in detail by prac- tical bankers before either can be trusted; but the point I wish to urge is, that some such special provision must be made to insure a return of the note-circulation to its former volume, or it will probably increase. In any bank note system, the whole volume of notes below the lowest point reached in the duller times is dead and useless for purposes of elasticity. It is only the flow and ebb that give the elasticity; if the notes never fall be- low 100 millions, that 100 millions might as well be coin.* We may accordingly, if we please, rid ourselves of practi- cally all the present ordinary note-circulation, substitute coin for it, and still have the desired elasticity by using the extra notes of one of the two plans described above. Of course the reduction ought to be made gradually, and by the weight of conditions that do not affect the extra notes, e. g., by a higher tax. OBJECTIONS TO STATE BANK NOTES. It is hard to see why the national bank system, as such, should be blamed for its failure to be elastic. Any state bank system must be subject to the same natural laws. £he transfer of supervision to the State governments makes no change whatever in the rule that the increase or de- crease of note-circulation will be determined by the excess of the profits over the checks or of the checks over the profits; that rule is, from the nature of the case, unchange- * Except that the change might introduce some additional difficulty into the arrangements for putting good security behind the extra notes, and the notes might become unfamiliar enough to be a little less convenient as money. 70 Colorado College Studies. able. All that can be done is to vary the nature of the checks; and wherever that can be done by the state gov- ernments it can be done by the national government. If it is desired to increase the note-circulation, that end is reached as certainly by removing taxes from national bank circulation as by removing them from state bank circula- tion. If it is desired to make temporary contractions easier, by reducing the expense and trouble of withdrawing de- posited bonds and redeeming the notes, or to adopt the Knox plan or the second plan described in this article, every detail of the improved process can be applied by the na- tional government as easily as by the state governments, and indeed with less expense. Anything that would be dangerous in a national bank note system would be as dangerous (and usually more dangerous) in a state bank note system. And any imaginable gain under state super- vision can be had as well or better under national super- vision. The difficulty of framing a good system of bank note law and supervision by forty-four distinct legislatures needs no detailed discussion, unless by a competent humorist. There is no hope that a jumble of various state-systems would “ average up” into a good bank-note circulation. If in ten states, five states, one state, the system is bad, there will appear bank notes imperfectly guarded by secu- rities or dependent upon too small a reserve, in which case it is only a question of time when some bank will let its notes go to protest; to say nothing of the moral certainty that the wish of the legislatures to see the new system well started would cause the checks upon profits to be set too low, so that (taking the country through) there would b<£ a large unnecessary original issue of notes, coming beforfi any need of trade called for them, and merely crowding out better money.* In whatever states the notes were insecure, those states would suffer annoyance and occa- sional loss by circulation of such notes; some of the notes * This point has not received the attention it deserves. Its practical impor- tance is great. State Bank Notes. 71 would pass into other states and carry annoyance with them; and the repute of good notes would be injured out- side of the state of their origin, because many men would not take the trouble to remember which states were sound, and would look with distrust upon any notes of a distant state, perhaps of any state but their own. Unless all the states agreed upon the same paper and similar printing, the detection of counterfeits would once more become a fine art, and the unskilled man and woman would be daily exposed to loss. It is not enough to answer that there is no danger of the return of the absurd and dangerous bank money of the first half of the century; if there is any failure to redeem notes, any discount upon some notes of distant origin or of tarnished fame, the opportunity of petty fraud upon the ignorant and careless is thrown wide open, and every man must choose between possible loss and the vexatious precaution of examining every note he receives. Moreover, the workman in taking wages, the retail dealer in taking payment from a customer he is anxious to retain, is often reluctant to give offense by ob- jecting to money on the mere chance that it is depreciated, and will sometimes take it when he knows there will be a small depreciation, rather than raise the objection. The presence in any money system of elements that are dis- trusted, that must be looked for and chaffered over or sub- ject the recipient to loss, is so wearisome an addition to the friction of trade that a clear case of beneficence in other directions (and no small beneficence) must be shown to give such kinds of money any claim to consideration. And when such beneficence as state bank notes are capa- ble of can be had without the friction and annoyance by using national bank notes instead, the attempt to substi- tute the former seems a ludicrous folly. Even if we im- agine the prodigy of wise concurrent action of all the states at first, what guaranty is there against the appear- ance from time to time hereafter of those legislatures whose pride it is to despise experience, to brush aside the sophis- tries of prejudiced conservatives, and to open short cuts to 72 Colorado College Studies. prosperity for the oppressed plain people? Many of the advocates of restoration of state bank notes see the diffi- culties distinctly enough to propose that there shall still be some national restriction and supervision. The best that can be said of such plans is that a safe built partly of iron and partly of wood will resist fire better than a safe built wholly of wood. The advocates of an admixture of wood may fairly be asked what good is expected from the change. If we are to attempt elasticity, then, common prudence requires the preference of national bank notes to state bank notes; for any tolerable system of so-called state bank notes must include national control so nearly complete that it is really a national bank note system. III.— INCREASE OF SCANTY LOCAL CIRCULATION. So far as it deals with the increased demand at crop- moving time, this argument for state bank notes is of course mere repetition of the argument from elasticity; but a separate place may be given to it because some persons have seriously urged that the permanent stock of money could thus be increased in the regions where a scarcity is now felt. Taken in this sense, even, it is completely an- swered by the answer to the elasticity argument; for if there is any way by which state bank notes could give a larger permanent money circulation to parts of the country, the same way can be provided for national bank notes by national law and supervision. And if this were not so, the proposed advantage of a larger local circulation is im- aginary. Money is sent away from districts where capital is scanty because goods are so valuable to the farmer, the trader and the manufacturer, that they prefer having more goods to keeping a comfortable cash reserve in hand or in bank; indeed they usually borrow besides. Double the money in such a district, and within two months there would be little more than at first, if it were money that could be used in outside payments; in the long run there State Bank Notes. 73 would scarcely be more, unless it depreciated, than if the addition had never been made. A district within a coun- try can no more enlarge its permanent circulation by issu- ing paper than the country itself can; whatever nominal enlargement may appear will be caused by depreciation. Even a nominal permanent enlargement could be had only through notes that were objectionable outside and were issued in such volume as to exceed* the formerly normal amount of money in the district; until they reached that volume they would merely displace other money. With a circulation thus enlarged, the district would enjoy all the blessings of a depreciated paper currency and a varying rate of exchange with the outside world; and unless the district had a well defined frontier line, which is improb- able, there would be a strip of country all around it where the notes would be the cause of daily disputes and frauds. If we want this state of things, we can have it; but we can give the local banks free rein just as easily by national law as by state law. IV.— EVILS OF THE TEH PEE CENT. TAX. The objection to the tax on state bank circulation ap- pears in two forms. One of these, urging that individual liberty is unfairly dealt with by a prohibitory tax on bank notes that fail to conform to legally set rules, must be the product of hasty writing without second thought. Surely the creation of money is, like marriage, “affected by a public interest,” and subject to the right of the whole community to protect its interest by prescribing conditions for every such transaction. Does any one (except those who follow their truth wherever it leads, with a Tolstoi- like disregard of consequences and of other truths) pro- pose that every mushroom bank or factory store shall be left free to circulate its notes or “orders” as money where it can? If not, some power must prohibit, and prac- * Strictly, of course, allowance must be made for the scattered remains of the former money, and conversely for the increasing needs of trade. And to cause the inflation the notes must become practically inconvertible. 74 Colorado College Studies. tically the power must be either the Nation or the State. Here rises the other form of the argument: that the pro- hibitory tax goes beyond the constitutional powers of Con- gress. Such a claim can be maintained only as a protest against the decision of the Supreme Court, like the still existing dissent from the Legal Tender decision. There is no room for the claim of “moral” unconstitutionality, such as there might have been if the Court had merely sustained the law in its formal aspect of laying a tax, for the decision was not reached by ignoring the prohibitory nature of the tax, but by affirming the power to prohibit. The words are: “Congress may restrain by suitable enact- ments the circulation as money of any notes not issued under its own authority.” Ought Congress to retreat be- fore constitutional scruples which have (to state the case mildly) nothing near unanimity of legal opinion to sup- port them and have already been overruled by the Supreme Court, scruples 'which at best affect merely the question whether Congress or the State legislatures shall use a power that one or the other must use; when the abandonment of the field by Congress would expose the country to the in- evitable evils of disjointed management of note-circulation by the separate States? If any one holds that decision of the Supreme Court to be erroneous, he, might better aim to cure the error by a constitutional amendment than to put upon the country the needless and ridiculous embar- rassments that state bank paper would inflict. Between national bank notes and state bank notes, then, the choice seems too easy to be called a problem. The real bank-note question is whether we need any bank notes at all; and if we do, how, under national control, they can be kept secure and their volume can be made elastic. The latter part of the subject has had a fairly full public dis- cussion, but too little attention has been given to the pos- sibility of doing without bank notes. That the withdrawal of the present stock of bank notes would not probably be State Bank Notes. 75 a troublesome process, has been shown in the earlier part of this paper. And if it is thought necessary to secure for the money-supply an elasticity of volume, in addition to the existing virtual elasticity by change of speed in circu- lation and by additions of metal, it is possible to have elasticity without notes. Our choice of means is not restricted to notes. The Treasury itself is a ready-made apparatus. Whenever Treasury receipts exceed Treasury payments, the circula- tion outside contracts;* wdienever the payments predomi- nate, the circulation enlarges. When such a change comes without any corresponding need of trade, the disparity is set right by the usual automatic methods, i. e. by a quick- ening of the average speed of circulation or an increase of bank reserves in the cases of needless contraction and ex- pansion respectively, followed (if the disparity is large enough and lasts long enough) by a change in the rate of short-time interest and ultimately by a change in the ex- port or import of metal. When the expansion or contrac- tion happens to fit a need of trade, we have a true elasticity imparted by the Treasury action. In fact, several impor- tant outpours and. absorptions of money by the Treasury occur every year; if they could be made to fit the needs of trade better than they do now, the circulation would be made elastic pro tanto. For instance, nearly all the in- terest on the public debt is paid in April and the corre- sponding quarter months, now that the interest-bearing debt is mostly the 1907 four per cents. If the interest were made semi-annual, and payable in March and September, about 11 millions f of interest-money would be cast into the channels of business at crop-moving time, and another 11 millions in time for the spring trade, while there would be a corresponding contracting influence at work through * Contracts relatively ; that is, if greater forces are just then expanding it, Treasury action merely diminishes the expansion. The word “ enlarges,” in the next phrase, is used in the same relative sense, though the enlargement is usually absolute also. t Instead of the present arrangement of 514 millions in October, and 514 more in January, when it is not wanted. 76 Colorado College Studies. the summer and the winter, when it is wanted. Again, the pensions are paid quarterly, but are distributed in groups upon different sets of quarter-days. These might be so grouped, probably without serious inconvenience to the pensioners,* that the payments in two months of spring should be about twice as large and in two months of autumn about three times as large as at other times. With pen- sions amounting to 160 or 180 millions annually, such an arrangement could be made to “plump” an extra 30 or 40 millions in the autumn and half of that in the spring. Without recommending these particular methods, one may take them as illustrating the possibility of such adjust- ments. Other means, some of which are applied by the Treasury, are the purchase of bonds at the end of summer, the increase of Treasury deposits with the banks, the ar- rangement of contracts so as to make the principal pay- ments for supplies in the autumn and spring. It is a fair question for debate whether this paternal- looking behavior of the Treasury would be good policy in the long run. The objection to it as government inter- ference is sound. But as compared with the present Treas- ury methods, it would be only a better arrangement of ebbs and flows that exist already in considerable degree and often at the wrong times.f Whatever evils it has, the Treasury method of obtaining elasticity is perhaps better than any known bank-note system, and is far better than any possible state bank system. In the long run, among fallible men, expensive money is the most economical; money that cannot be duplicated without a liberal expense of labor; metal, or certificates known to have their full equivalent of metal behind them. In the sacrifice a coun- try necessarily makes to obtain such money, as compared *Crudel/, by making a part of the pensions semi-annual, with the autumnal payment larger than the other ; an inequality that fits tolerably well the needs of indigent pensioners. fSee illustrative figures in Mr. Kinley’s article on the Independent Treasury in the Annals of the American Academy of Political and Social Science , Septem- ber, 1892. The figures are for operations at New York, but are a sufficient indica- tion of what happens in the whole country. State Bank Notes. 77 with watering it by a note-element, it is buying business security and a charm against popular delusions. Some- thing might be said, even if no project for elasticity through Treasury payments is accepted, for the belief that our best policy is to prefer safety to elasticity, to arrange the checks on profit so as to insure the gradual extinction of the bank notes,* to substitute nothing for them, and so let coin flow in; when that is done, to increase the coin reserve against greenbacks by 15 or 20 millions annually till it equals the whole volume of greenbacks and they can be converted into coin certificates. When a gen- eration has grown up that has never seen a bank note or a government note, the seed of many a folly will be dead in the popular mind. The principal conclusions we have reached may be summarized as follows: 1. The extinction of the national bank notes requires no creation of other money. The coin to take their place will come in without perceptibly greater national sacrifice than has attended the extinction of a still larger amount of notes in the past few years. 2. An elastic bank note system would probably cause progressive increase in the amount of note-circulation, unless restraints hitherto unusual were applied. Such re- straint might be given by causing the notes added in the temporary expansions to bear interest after a set time, or by expressly requiring the issuing banks to reduce their circulation within the old limits before a set time. 3. Nothing is to be gained, either towards elasticity or towards permanent enlargement of local circulation, by substitution of state control for national control. The in- crease or diminution of bank notes is determined by the relation between the profits and certain well defined checks, and the state governments cannot apply the checks better than the national government can. * Which of course does not imply a disappearance of the national bank system. 78 Colorado College Studies. 4. The relegation of control to the states would almost certainly cause the checks to be set too low at first, and cause a large initial increase of notes. 5. Insecure notes from a single state would diminish the practical convenience of the whole note-circulation. 6. Security of all the notes under separate state man- agement is almost incredibly improbable; and if attained in one year, might easily fail in the next. 7. The prohibition, by tax or otherwise, of circulating notes that fail to conform with national law, is not an in- justice to individuals. The objection on constitutional grounds touches only the question whether state or na- tional legislatures shall impose the necessary restrictions; and if sound at all, is rather an argument for a constitu- tional amendment than for a bad money system. 8. A means of elasticity, probably safer than any bank note system, exists in the irregularity of Treasury expenses. 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