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202 A GENERAL CONSTRUCTION FOR CIRCULAR CUBICS. [May, 

ing: "In the liguric documents of the second half of the fifteenth century we 
found in frequent use, to indicate the multiplication by 1000, in place of M, an 
O crossed by a horizontal line." This closely resembles some forms of our 
Spanish symbol U. Cappelli gives two facsimile reproductions 1 in which the 
sign in question is small and is placed in the position of an exponent to the letters 
XL, to represent the number 40,000. This corresponds to the use of a small 
c which has been found written to the right of and above the letters XI, to signify 
1100., It follows, therefore, that the modified U was in use during the fifteenth 
century in Italy, as well as in Spain, though it is not known which country had 
the priority. 

What is the origin of this calderonl Our studies along this line make it 
almost certain that it is a modification of one of the Roman symbols for 1000. 
Besides M, the Romans used for 1000 the symbols c|d, T, °° and *f\ These 
symbols are found also in Spanish MSS. It is easy to see how in the hands of 
successive generations of amanuenses, some of these might assume the forms of 
the calderon. If the lower parts of the parentheses in the forms c | d or c 11 d 
are united, we have a close imitation of the U, crossed by one or by two bars. 

Allied to the distorted Spanish U is the Portuguese symbol for 1000, called 
the " Cifrao." 2 It looks somewhat like our modern dollar mark $. But its 
function in writing numbers was identical with that of the calderon. Moreover, 
we have seen forms of this Spanish "thousand" which need only to be turned 
through a right angle to appear like the Portuguese symbol for 1000. Changes of 
that sort are not unknown. For instance, the Arabic numeral 5 appears upside 
down in some Spanish books and manuscripts as late as the eighteenth and 
nineteenth centuries. 

A GENERAL CONSTRUCTION FOR CIRCULAR CUBICS. 

By R. M. MATHEWS, Wesleyan University. 

1. It is well known that any cubic curve can be generated as the locus of the 
intersections of a pencil of conies on four points with a projective pencil of lines. 
In practical work it is difficult to draw the conies and to effect the correlation 
involved in this construction. As a circular cubic passes through the circular 
points at infinity, the conies for such a curve may be specialized to a pencil of 
circles on two finite points. It remains, then, to find a simple and general 
method for effecting the correlation with the pencil of lines. 

Schroeter and Durege in simultaneous papers 1 have shown that if each line 
pass through the center of the corresponding circle, the locus contains its singular 

1 Adriano Cappelli, op. cit., p. 436, first column, Nos. 5 and 6. 

2 See the word "Cifrao" in Antonio de Moraes Silva, Dice, de Lingua Portuguesa, 1877; 
in Vieira, Grande Dice. Portuguez, 1873; in Dice. Comtemp. da Lingua Portuguesa, 1881. 

'H. Schroter, "Ueber eine besondere Curve Z ur Ordnung und eine einfache Erzeugungsart 
der allgenieinen Curve 3 tir Ordnung," Mathematische Annalen, vol. 5, 1872, pp. 50-82. 

H. Durege, "Ueber die Curve 3" r Ordnung, welche den geometrischen Ort der Brennpunkte 
einer Kegelschnittschaar bildet," ibid., 83-94. 




1922.] A GENERAL CONSTRUCTION FOR CIRCULAR CUBICS. 203 

focus, which is the intersection of the tangents at the circular points. Moreover, 
the general construction has been considered for special base points 1 . This 
paper explains a general and simple compass and ruler construction for the 
case of an arbitrary pair of base points. 

2. Let Q and R be two fixed points on a variable circle which is cut again in 
S and T by two arbitrary fixed lines s and t through Q and R, respectively. The 
variable line ST cuts an arbitrary fixed line g in U. 
The variable line PU, drawn from a fixed point P, 
cuts the circle in A and B. The locus of A and B for 
the pencil of circles through Q and R is a cubic curve. 
We find the following properties of the figure. 

1. The lines ST are parallel, for z TSQ = Z QRT 
= constant. 

2. There is a one-to-one correspondence between Fig. *• 

the circles and the pencil at P. For each circle cuts s and t in points S and T; 
the line ST determines U on g and so PU. Conversely, each line from P cuts g 
in a point U which determines one line in the fixed direction ST and so two points 
S and T concyclic with Q and R. 

3. Q and R are on the locus. For PQ cuts g at U q and so the corresponding 
circle is determined; a similar consideration applies to R. 

4. The circles of the pencil pass through the circular points at infinity, and 
thus the isotropic lines through P place them on the locus, which is, then, a 
circular cubic. 

5. The circle through P puts this point on the curve. 

6. When U is at infinity on g, then S and T are at infinity, and the circle on 
Q and R is replaced by the line QR and the line at infinity. Let the line through 
P parallel to g cut QR at P' and the line at infinity at F a , which is also on g. 

7. The line g cuts the locus at F M and at the points U s and U t where it meets 
s and t, respectively. 

3. The given construction can now be shown applicable to any circular 
cubic. A circle through three points Q, R and A of the curve will cut it again in 
a finite point B. The line QR cuts the curve again in P' while AB does so in P. 
Then it is well known that PP' is parallel to the real asymptote. Let g, an arbi- 
trary parallel to PP', cut the cubic in U s and U t . Thus lines s and t are deter- 
mined as QU S and RU t , respectively. Our construction applied now on this 
basis will give a circular cubic which has in common with the given cubic the 
nine points, Q, R, P', A, B, P, F K , U s and U t besides the circular points at 
infinity. Accordingly, the cubics are identical. 

4. The construction which Teixeira 2 derived analytically may be shown to 
be a special case of the foregoing. Take g through R and let it cut the cubic 

1 G. Loria, Spezielle algebraische und transzendente ebene Kurven. Leipzig, 1910, vol. 1, pp. 
34-35. 

2 F. Gomes Teixeira, "Sur une maniere de construire les cubiques circulaires," Nouvelles 
Annales de Mathematiques, fourth series, vol. 16, 1916, pp. 449-454. 



204 



A GENERALIZATION OF THE STROPHOID. 



[May, 



again at U,. Let the variable circle cut g again at H. Then as triangles QRU S 
and SHU S are similar: 

U*H_U S Q 

U S S UsR 




Fig. 2. 



k, constant. 



As the variable line i$T moves parallel to itself, 
the segments it cuts on the fixed lines from U s are 
proportional, so 



Thence 



=rr— = k , constant. 






k_ 
k' 



c, constant. 



The procedure given is to draw PU arbitrarily from P to cut g at U; to determine 
H on g from the constant ratio just proved; and then to construct the circle 
through QRH to cut PU in points of the curve. 



A GENEEALIZATION OF THE STROPHOID. 

By J. H. WEAVER, Ohio State University. 

W. W. Johnson has given the following generalization for the strophoid. 1 
Let A and B be two fixed points, and let two variable lines PA and PB make 
with AB angles <j> and \f/, respectively. Let a be a constant angle and let P 
move so that 

n<f> ± rrup = a. (1) 

Then the locus of P is a strophoid. Equation (1) shows that there is associated 
with this set of curves a circle having a segment with AB as base in which the 
angle a may be inscribed. 

In the following discussion some curves are developed which have associated 
with them the three conic sections. 

Elliptic Case. Let there be an ellipse E with major axis AB, and from 
A and B let variable lines AP and BP be drawn making angles di and 2 respec- 
tively with AB. Let AQ and BQ be so drawn as to make angles ± mdi and 
± ndi with AB. When the locus of Q is the ellipse E, the locus of P is a curve 
whose equation may be developed as follows. (In this development we will 
consider m and n as positive integers and relatively prime to each other.) 

The slope of AQ is tan (m6\) and of BQ is tan (w0 2 ). Then since Q is on E 
we have 

tan (mOi) -tan (nd 2 ) = - b 2 /a 2 , (2) 

1 "The Strophoids," American Journal of Mathematics, vol. 3, 1880, pp. 320-325. See also 
G. Loria, Spezielle algebraische und transcendente ebene Kurven, Berlin, vol. 1, 1910, p. 73. This 
Icass of curves includes the sextrix curves as a subclass. See Loria, I.e., p. 390.