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Buntingford in Hertfordshire, with the characters 21^ 1 6 ; which had been read
10l6, supposed to be mixed numbers, the 319 Roman, ^"d the two others
Arabian or Indian, as they are indifferently called. This led him to consider
two other dates of the like kind, formerly published in the Philos. Trans. ; one
found at Helmdon in Northamptonshire, in mixed characters expressing, as
was thought, iH^ '33; and the other at Colchester, said to denote the year
lopo, wholly in Arabian figures. But on searching into the origin of those
figures, and the time when they were first brought into these parts of the
world, he could meet with no examples of them in any manuscripts, before
some copies of Johannes de Sacro Bosco, mentioned by Dr. Wallis, who died
in the year 1256, which was 123 years after the latest of the three dates above-
mentioned. As it could not therefore but seem very strange, that workmen
should have made use of those figures for such common purposes, so long be-
fore they appear in the writings of the learned ; so on a closer examination,
and further inquiry, he found there was no reason, from any of these dates, to
suppose it was really true in fact. For the Helmdon date, instead of M) 133,
should be read 21^233; the Colchester date 14gO, instead of JO9O; and that
at Widgel-Hall has no Arabian figures in it, the characters 1 and 6 not being
numbers, but the initial letters of two proper names I G, in the usual form of
those letters in that age.

But there had been soon after read before the Society, an account of a date
at Worcester, more ancient than any of the 3 former; namely OtIOj or 97^,
in which the unit is a Roman numeral, and the other two are taken for Indian
figures. Now Mr. Ward observed in his former paper, that such mixtures
were sometimes found in ancient numbers; though in what manner they were
so used, he did not then explain, but for brevity contented himself with refer-
ring to Dr. Wallis's Algebra. The Doctor thought it necessary to take notice
of this, in order to account for his way of reading the Helmdon date, in which
the a? only is a Roman numeral. And Mr. Ward had met with a few instances
of it in Dr. Mead's manuscript of Boethius, as 00029 'Jnd docSs, where the
hundreds are numeral letters, and both the decimals and units Arabian figures,
(De Arith. lib, ii.) But it is observable, that this is not done promiscuously ;
for the larger numbers are always letters, and the less figures ; as in the Helm-



40 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735.

don date. And MabUlon has observed, that in a curious manuscript copy of
Thomas k Kempis, written in the 15th century, some of the pages are so
numbered, (De Re Diplom. tab. xv.) Which method, so far as appears, was
always attended to, and never in any one instance inverted. So that this
Worcester date, which has a Roman numeral in the place of units, and the
two preceding characters are supposed to be Indian figures, is not only without
example, but directly contrary to all other instances of such mixed numbers.
Which consideration alone might be of sufficient ground to think, there must
be some mistake in the reading.

But the middle figure, taken for a seven, is as remarkable ; which turning
towards the left hand, forms two obtuse angles, one above, and the other be-
low. This shape of the seven was never seen before, and seems by no means
to suit that age. In the specimen of the figures taken from Johannes de Sacro
Bosco, by Dr. Wallis, the figure seven is made in this form A, like the two
legs of an isosceles triangle. And in Roger Bacon's Calendar, dated 12Q2,
there is only this variation, that the leg to the left hand is somewhat shortened.
And this form continued till printing was introduced among us ; as is evident
from Caxon's Polychronicon, and other books printed about that time. Nor is
it found till later times in any other shape ; unless that in Bishop Beveridge's
table of Indian figures, the two legs of our ancient seven are drawn parallel, and
arched at the top, instead of meeting in an angle; (Arith. Chron. lib. i, cap. 4.)
and Planudes, a Greek writer, has kept the true Arabian form V, like the Ro-
man five, which the Europeans inverted. The last alteration this figure re-
ceived among us, was by raising the shorter leg horizontally. But no instance
of it parallel to this in the Worcester date, or any thing like it, has before ap-
peared. As there seems therefore no reason to suppose it a seven ; so a proba-
ble conjecture may be offered, what it was designed for, and that is, the Roman
numeral ten, which was made in this form, like an X ; to which character,
in our old square hand, this supposed seven ^ would very well agree, by sup-
plying only the two extreme parts to the right hand, in this manner X, which
may easily be thought to have been decayed, and worn away by length of time.

As there is no reason to take the middle character for seven, so neither is
there any to suppose the first was intended for a nine, being thus placed before
two Roman numerals, as Mr. Ward takes them both to be. It has indeed some
similitude with that figure ; but that is nothing more than what was anciently,
and still is, common to the letter OQ in that hand, which resembles a double Q,
with an oblique stroke turned inwards from the bottom of that to the right
hand ; so that if the other to the left be taken away, that which remains will
appear in this form ^ > I'ke what is here called a nine. And every one knows.



VOL. XXXIX.J PHILOSOPHICAL TKANSACTIONS. 41

who has any acquaintance with ancient inscriptions, that letters frequently
perisl) in this manner, one part before another.

Upon these suppositions the true reading would be mxv. But since the old
date is now destroyed, and modern figures put in its place, this must remain
uncertain. But tiiough the precise year of tliis date cannot now be determined
with certainty, it is sufficient to have shown, that neither the order of the
characters, their shape, nor the oldest examples of Arabian or Indian figures,
any where found, do in the least countenance the reading given to it; but,
on the contrary, all of them afford the highest probability, that it cannot be
genuine.

On the Description of Curve Lines. By Mr. Colin Maclaurin, Math. Prof.
Edinb. F.R.S. N° 439, P- 143.

Mr. Maclaurin was informed that some papers had lately been presented
to the Royal Society, concerning the description of curves, in a manner that
has a near affinity to that which he communicated to them formerly, and had
carried farther since; and that it would not be unseasonable, nor unacceptable,
if he should send an account of what he had done further on that subject since
the year 17 IQ. The author* of those papers taught mathematics at Edin-
burgh privately for some years, and some time ago, viz. in 1727» mentioned
to Mr. M. some theorems he had on that subject; which, at the same time,
Mr. M. showed him in his papers. Some time before that, he showed him a
theorem which coincided with one of those in Mr. M.'s book, though he
seemed not to have observed that coincidence ; and indeed methods of that
kind, are often found coincident that do not appear such at first sight. Mr.
M. is unwilling to be the occasion of discouraging any thing that is truly inge-
nious, and renounces any pretensions of appropriating subjects to himself; but
on the contrary, wishes justice may be done to every person, or to any per-
formance in proportion to its merit ; yet finding it fit he should take precautions,
lest any one should take it in his head afterwards to say, he takes things from
him which he may have had long before him; and therefore Mr. M. sends the
following abstract of what he had done in this matter since the year 1719.

Mr. M. has so much on this subject by him, that he declares himself at a
loss what to send; but at present he only gives an abstract of those propositions,
which he takes to be more nearly related to those which this author has offered
to the society from the conversations he had with him. In 1721, Mr. M.
printed several sheets of a supplement to his book on the description of curve

* Mr. Braikenridge. See p. 5, of this volume.
VOL. VIII. G



42 I'HILOSOPHICAL TRANSACTIONS. [aNNO 1735.

lines, which he hud not published, having been engaged for the most part in
business of a different nature, and in pursuits on other subjects since that time.
He first gives an abstract of that supplement, as far as it was then printed, and
subjoins an account of some theorems he added to it the following year, viz.
in 1722. He was led into those new theorems by Mr. Robert Simson's giving
him at that time a hint of the ingenious paper, which has been since published
in the Philosophical Transactions. Mr. M. had tried, in the year 17 ig, what
could be done by the rotation of angles on more than two poles; and had ob-
served, that if the intersections of the legs of the jingles were carried oyer
right lines, as in Sir Isaac Newton's description, the dimensions of the curve
were not raised by this increase of the number of poles, angles, and right lines;
and therefore he neglected this at that time, as of no use to him, confining
hiuiself to two poles only, and varying the motions of the angles as in his
book. He found this by inquiring in how many points the locus could cut a
right line drawn in its plane, and found, by a method often used in his book,
that it could meet it in two points only.

Having found then, that three or more poles, were of no more service than
two, while the intersections were carried over fixed right lines; he thought it
needless to prosecute that matter then, since by increasing the number of poles,
his descriptions would become more complex, without any advantage. But in
June or July, J 722, on the hint he got from Mr. Simson of Pappus's porisms,
he saw that what he has there ingeniously demonstrated, might be considered
as a case of the abovementioned description of a conic section, by the rotation
of any number of angles about as many poles; the intersections of their legs in
the mean time being carried over fixed right lines, excepting that of two of
them which describes the locus. For by substituting right lines instead of the
angles, in certain situations of the poles and of the fixed right lines, the locus
becomes a right line; as for example, in the case of three poles, when these
three are in one right line, in which case the locus is a right line, which is a
case of the porism.

It was this that led him to consider this subject anew; and first he demon-
strated the locus to be a conic section algebraically ; and found theorems for
drawing tangents to it, and determining its asymptotes. He also drew from it
at that time a method of describing a conic section through five given points.*
This encouraged him to substitute curves for the right lines, to see if by this
method he could be enabled to carry on his theorems, about the descriptions of

* The paper on this subject I have, says Mr. M. is dated July 31, 1722, at sea, being then in my
way to London, going for Cambray. Orig.



VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 43

lines through given points, to the higher orders of lines. Some of the tiieo-
rems he ft)un(l at that time accompany this. In November 1722, looking into
Sir Isaac's Principia, he saw that the description of the conic sections by three
riglit lines, moving as above, about three poles, could be immediately drawn
from his 20th lemma, which itself is a case of this description. This gradually
led him to seek geometrical demonstrations for the whole, as far as it related to
the conic sections. He sent some leaves of this paper, dated at Nancy, No-
vember 17 22. Since that time, he had not added much to this subject, but
what relates to drawing tangents, determining the asymptotes, and the puncta
duplica, or multiplicia of these curves. He considered it the less, as he did
not find it more advantageous in any respect, than the method he had consi-
dered in tiis book, nor more general.

In 1727 lie added to a chapter in his algebra, an algebraic demonstration of
the locus, when three poles are employed; and the method of describing a
conic section through five given points, subjoining at the same time, that if
more poles are employed, and angles or right lines, the locus was still a conic
section ; which he thought was a remarkable property of the conic sections, not
observed before.

These things he intended to put in order, and publish in the supplement to
his book, a part of which had been printed since the year 1 721. He intended
also to give several other things in that supplement; two of which he only just
mentions at present, as they are foreign to the present affair. He subjoins a
problem determining the figure of a fluid, whose parts are supposed to be attracted
to two or more centres; and a solution of a general problem about the collision
of bodies.

The author of the papers given in to the Royal Society will not deny, that
Mr. M. showed him the theorems, now sent, in 1727. He owned it last sum-
mer at least; Mr. M. intended to publish these very soon. Whether he has
carried the subject farther, he leaves to the judgment of the gentlemen to whom
they were referred. As to the demonstrations, it would take some time to put
them in a proper form to be published. He could send those that are algebraic
easily; but did not care to send those that are geometrical, till more leisure.

An Abstract of ivhat has been printed since the Year 1721, as a Supplement to a
Treatise concerning the Description of Curve Lines, published in 1719, and
of what the ^-Juuhor proposes to add to that Supplement. By Mr. Maclaurin.
N°439, p. 148.

I. In the first part of the supplement, a general demonstration is given of
the theorem, that if two lines of the orders or dimensions, expressed by the

G 2



44 PHILOSOPHICAL TRANSACTIONS. [aNNO J 735.

numbers m and n, be described in the same plane, the greatest number of points
in which these lines can intersect each other, will be mn, or the product of
the numbers which express the dimensions of the lines, or the orders to which
they belong.

II. In the next part, theorems are given for drawing tangents to all the curves
that were described in that treatise by the motions of angles on given lines.
Their asymptotes are also determined by more simple constructions than those
which are subjoined to their descriptions in that treatise. Of these we shall
give one instance here.

Suppose the invariable angles, fig. 1 and 2, pi. 3, pcg, ksh, to revolve about
the fixed points or poles, c and s. Suppose the intersection of the two sides
OF, SK, to be carried over the curve bom, whose tangent at the point q is sup-
posed to be the right line ae ; and let it be required to draw a tangent at p to
the curve line described by p the intersection of the other two sides cg and sh.

Construction. — Draw qt constituting the angle saT, equal to caA, on the
opposite side of s«, that aA is from cq; and let qt meet cs, produced if neces-
sary in t. Join pt, and constitute the angle cpn equal to spt, on the opposite
side of cp, that pt is from sp; then the right line pn shall be a tangent at p,
to the curve described by the motion of p, which is always supposed to be the
intersection of cg and sh.

The asymptotes of the curve, described by p. are determined thus. Find,
as in the abovementioned treatise, when these sides become parallel, whose
intersection is supposed to trace the curve; which always happens when the angle
cas becomes equal to the supplement of the sum of the invariable angles fcg,
KSH, to four right ones; because the angle cps then vanishes. Suppose, in
fig. 3 and 4, that when this happens, the intersection of the sides of, sk is
found in a.

Constitute the angle saT equal to caA, as before, and let aT meet cs in t.
Take cn equal to st, the opposite way from c that st lies from s. Through n
draw DN parallel to cg or sh, which are now parallel to each other; then dn
shall be an asymptote of the curve described by the motion of p.

If instead of a curve line bqm, a fixed right line ae be substituted, then the
point p will describe a conic section, whose tangents and asymptotes are deter-
mined by these constructions. In this supplement, it is afterwards shown how
to draw the tangents and asymptotes of all the curves which are described in
the abovementioned treatise by more angles and lines.

III. The same method is afterwards applied to draw tangents to lines described
by other motions than those which are considered in that treatise ; of which the
following is an instance. Suppose that the lines CP and sp, fig. 5, revolve



VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 45

about the poles c and s, so that the angle acp bears always the same invari-
able proportion to asp, suppose that of m to n. In the line cs, take the point
T, so that ST may be to ct in that same proportion of ot to n; then this point
T will be an invariable point; sin(;e cs is to ct. as ot — n to n. Draw tp, and
constitute the angle spn, equal to cpt, so that pn and pt may lie contrary
ways from sp and cp, and pn shall be a tangent of the curve described by
the motion of the point p. Several other theorems of this kind are subjoined
here.

IV. After these, lines or angles are supposed to revolve about three or more
poles, and the dimensions of the curves with their tangents and asymptotes are
determined. Suppose in the first place, that the three poles are c, s, and d,
fig. 6, and that lines or rulers ck, sq, qdr, revolve about these poles. The
line which revolves about d, serves only to guide the motion of the other two,
so that its intersection with each of them being carried over a fixed right line,
their intersection with each other describes the locus, which is shown to be a
conic section. The intersection of qdr with sq, is supposed to be carried over
the fixed right line af ; the intersection of the same auR with cr, is supposed
to be carried over the fixed right line ae ; and in the mean time, the intersec-
tion of the right lines sa, cr, that revolve about the poles s and c, describes a
conic section.

This conic section passes through the poles c and s; and if you produce dc
and DS, till they meet with Aa and ar in f and e, it will also pass through f
and e: it also passes always through a the intersection of the fixed lines qf
and ER; from which this easy method follows, for drawing a conic section
through five given points. Suppose that these five given points are a, f, c, s,
and e: join four of them by the lines af, fc, ae, es, and produce two of these
FC, es, till they meet, and by their intersection give the point d. Suppose
infinite right lines to revolve about this point d, and the points c and s, two
of those that were given, and let the intersections of the line revolving about
D, with those that revolve about c and s, be carried over the given right lines
AE, af; then the intersection of those that revolve about c and s with each
other, will, in the mean time, describe a conic section, that shall pass through
the five given points a, f, c, s, and e.

It is then shown, that when c, s, and d are taken in the saine right line, the
point p describes a right line, fig. 7> as also when c, s, and a are in the same
right line; which also follows from what is demonstrated in that very ingenious
paper concerning Pappus's porisms, communicated by Mr. Simson, professor
of mathematics at Glasgow, published in the Phil. Trans. N" 377.

In the next place it is shown, that if four right lines revolve about four poles



46 PHILOSOPHICAL TRANSACTIONS. [aNNO 1735.

c, s, D, and e, and those that revolve about d and e, serve only to guide those
that revolve about c ands; so that a and h, the intersections of that which
revolves about d, with those that revolve about e and s, be carried over the
fixed lines ab and af; and m the intersection of that which revolves about k
with that which revolves about c, be carried over a third fixed line bp; then the
intersection p of those that revolve about c and s, will, in the mean time, de-
scribe a conic section, and not a curve of a higher order. The conic section
degenerates into right lines, when cp and sp coincide at the same time with the
line OS, that joins the poles c and s, as in the preceding description ; which
coincides again with what is demonstrated in the abovementioned ingenious
paper.

After this it is shown generally, that though the poles and lines revolving
about them be increased to any number, and the fixed lines over which such
intersections, as we described in the last two cases, are supposed to be carried,
be equally increased, the locus of the point p will never be higher than a conic
section; that is, let a polygon of any number of sides have all its angles, one
only excepted, carried over fixed right lines, and let each of its sides produced,
pass through a given point or pole, and that one angle, which we excepted, will
either describe a straight line, or conic section.

Thus, if a hexagonal figure lqrpmn, fig. 8, have all its angles, excepting p,
carried respectively over the fixed right lines Aa, sb, og, nh, kIc; then the
point p in the mean time will describe a conic section, or a right line. The
locus of p is a right line when cp and sp coincide together with the line cs.
All these things are demonstrated geometrically.

V. After this, angles are substituted instead of right lines revolving about
these poles; and it is still demonstrated geometrically, that the locus of p is a
conic section or right line.

Suppose that there are four poles c, s, d, and e, fig. 9, about which the in-
variable angles pca, psk, rdm, meu revolve; and that a, m, and r, the inter-
sections of the legs cq and eq, of em and dm, and of dr and sr, are carried
over the fixed right lines Aa, sb, Gg, respectively; then the locus of p is a conic
section, when cp and sp do not coincide at once with the line cs; but is a right
line when cp and sp coincide at the same time with cs; and never a curve of a
higher order.

VI. Having demonstrated this, which seems a remarkable property of the
conic sections, or lines of the second order; it proceeds to substitute curve
lines instead of right lines in these descriptions, as is always done in the treatise
concerning the description of lines, and to determine the dimensions of the
locus of p, and to show how to draw tangents to it to determine its asymptotes.



VO?.. XXXIX.] PHILOSOPHICAL TRANSACTIONS. 4ij

and other properties of it. Mr. Maclanrin had observed in 1719> that by in-
creasing tlie number of poles and angles beyond two, the dimensions of the
locus of p, did not rise above those of the lines of the second order, while
the intersections moved on right lines; and therefore he did not think it of use
then to take more poles than two, since by taking more, the descriptions be-
came more complex, without any advantage. When the intersections are car-
ried over curve lines, the dimensions of the locus of p rise higher, but the
curves described have double, or multiple points, as well as when two poles only
are assumed; and therefore this speculation is more curious than useful. How-
ever, he subjoins some of the theorems that he found on this subject, con-
cerning the dimensions of the locus of p, and the drawing tangents to it.

1. If, in fig. 6, you suppose q and r to be carried over curve lines, of the
dimensions m and n respectively; then the point p may describe a locus of Imn
dimensions.

2. If, in fig. 8, you suppose L, a, r, m, n, to be carried over curve lines of
the dimensions wi, n, r, s, t, respectively; then the locus of p may arise to
Qimnrst dimensions, but no higher; and if instead of lines revolving about
the poles, you use invariable angles, the dimensions of the locus of p will rise
no higher.

3. He then assumed 3 poles, c, d and s, (fig. 10) and supposed one of the
angles snl, to have its angular point n carried over the curve an, while the
leg NG passes always through s, as in the description in the treatise of the ge-
neral description of curve lines, while the angles odr, rcp, revolve about the
poles D and c : he supposes also the intersections a and r to be carried over the
curve lines sa, gr, and that the dimensions of the curve lines an, Ba, gr. are
m, n, r, respectively; and finds that the locus of p may be of Smnr dimensions;
but that the point c is such, that the curve passes through it as often as there
are units in 2mnr.

4. If any number of poles are assumed, so as to have angles revolving
about them, as about c and d in the last article, and the intersections are car-



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