Royal Society (Great Britain).

The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) online

. (page 46 of 85)
Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 46 of 85)
Font size
QR-code for this ebook


Hebrew, Phoenician and Egyptian etymologies, he often makes the names,
which at first sight are almost all quite unlike, to be the same in sound, or at
least in .sense. And by this application of his skill in the ancient languages,
he readily finds out a coincidence between Moses's and Sanchoniathon's earliest
generations.

But his main work, and what he appears most pleased with, is his discovery
of Abraham and his family, among the latter generations recorded by Sancho-
niathon. Having laid down, that Ouranos is Terah, the father of Abraham,
he undertakes to prove, that Abraham is the Chronus of Sanchoniathon, and
the Saturnus of the Latins ; that Sarah, his wife, is the same with the goddess
Rhea ; that Ishmael, Abraham's son, is the Miith of Sanchoniathon, and the
Dis or Pluto of the Greeks and Romans : that Isaac, Abraham's other son, is
the same with the Sadid of Sanchoniathon, with Jupiter among the Latins, and
Zjuf among the Greeks, his wife Rebecca being Juno ; that Esau, Isaac eldest
son, is Osiris and Bacchus ; and that Jacob, the youngest, is Typhon. And,
in like manner, he finds a very great part of the Grecian theology in Abraham's
family.

In the mean while his readers will, perhaps, make two very material obser-
vations on this extraordinary discovery of his : the one, that Chronus's cha-
racter in Sanchoniathon's fragment, is the most immoral and tyrannous of any
recorded there : and how to reconcile this with the character given in scripture
to Abraham, as the friend of God, the father of the faithful, &c. is no easy
task : it requires, to be sure, more than a resemblance of two or three circum-
stances, common to Chronus and Abraham, when their historians in fifty other
circumstances make their characters essentially different. The other con-
sideration, which occurs, when we read this treatise, is, that Abraham had ill
luck indeed, if, when he left his native country because of the rise of idolatry
there, all the grosser idolatry of the heathen nations after his time took its rise
from him and his family : the very crime which he took pains to avoid, he was
the accidental occasion of, if he and his are to be thus placed at the head of
the heathen theology.

The author, having finished this remarkable part of his work, enters into a
very learned detail of the particular Gods of the several heathen nations, who
are the most celebrated in history ; and he has shewed a great compass of



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 3J)(|

reading on this occasion. Hardly any writer has been more copious on the
subject, or has given better hints for clearing up many passages of sacred and
profance story.

In his third book he has treated at large on the dynasties of Egypt, and the
shepherd kings who reigned there : both of them, perhaps, the darkest spots
in the whole face of antiquity. He has taken great pains to fix the epochs of
the kings of Sicyon, Sidon and Tyre, of Arabia, Assyria, Lydia, of the Medes
and Babylonians ; concerning all which, he has laid together the most remark-
able testimonies of the ancients. At length he comes to his favourite point,
the Chinese history, and gives, as he says, a complete list of their kings,
from the flood down to the present monarch of that empire, and shows that
the chronology of the Chinese, may be made pretty nearly consistent with the
true chronology of the Old Testament.

And for this part of the work the author seems well fitted, being skilled, as
he tells us in the preface, in the learned characters of that country, which
he has studied for near 20 years, and has for some time taught in the royal
college at Paris ; and having composed 5 dictionaries and a grammar, of that
language, with a translation, almost intire, of the geography of Tamim, which
contains no less than the whole history of that empire : on which occasion he
applies to himself, and the progress he has made in the Chinese learning, those
expressive verses of Virgil in his 6th book of the ^neid :

— — ^^— Pauci, quos aequus amavit

Jupiter, aut ardens evexit ad aethera virtus,

Dii geniti, potuere.

On the Scurvy-Grass that grows in Greenland. By Mr. David Nicholson, Sur-
geon. N°45(), p. 317.

Mr. N. communicates this as matter of truth, and not hypothetic, viz. that
the scurvy-grass in Greenland, equally the same with ours in England, as to
the figure of the plant, and all its appearance to the eye, changes its nature
much, as it approaches the sun ; for in that climate, its principal quality, the
volatile salt, is neither pungent nor perceivable ; but to the taste, the whole
plant is quite as insipid as the colewort or beet. Mr. N. preserved some plants
with their natural earth, and brought them to London alive ; and he observed
the remarkable change produced by the sun's heat on them ; for the saline
matter in Greenland, which certainly was analogous to a fixed salt, became, in
a month's time, almost to the same volatility as that which naturally grows in
England.



392 PHILOSOPHICAL TRAKSACTIONS. [aNNO 1740.

Concerning two Species of Lines of the Third Order, not mentioned by Sir Isaac
Newton, nor by Mr. Stirling. By Mr. Edmund Stone *, F. R. S. N" 456
p. 318.

Mr. Stone having for some time past been reading and considerirtg the little
treatise of Sir Isaac Newton, intitled, Enumeratio Linearum tertii Ordinis, as
also the ingenious piece of Mr. Stirling, called, Illustratio Tractatus Domini
Newtoni Linearum tertii Ordinis ; he observed, that they have neither of them
taken notice of the two following species of lines of the third order; and he
ventures to affirm, that the 72 species mentioned by Sir Isaac, with the 4 more
of Mr. Stirling, and these 2, making in all 78, is the exact number of the dif-
ferent species of the lines of the 3d order, according to what Sir Isaac has
thought fit to constitute a different species.

The 2 species are to be reckoned among the hyperbolo-parabolical curves,
having one diameter, and one asymptote, at N° 8 of Newton's Treatise, or
p. 104 of Mr. Stirling's ; its equation being xyy = + bx'^ + ex + d; which will
give, not 4, as in these authors, but 6 species of these curves : for,

1. If the equation bx^ + ex -\- d ^ O, has 2 impossible roots, the equation

* Mr. Edmund Stone was a rennarkable instance of the effect of industry united to good natural
talents; having raised liimself to an eminent rank in the mathematical sciences, as well as in the
languages, by his own application alone ; all the instructions he ever received, being only to know
the 24 letters of the alphabet, which he was taught at 8 years of age, by a servant in the duke of
Argyle's family, where young Stone's father was gardener ; with whom, also, at an early age, the
son became a servant ; in which situation he spent a considerable part of his life. When about 18
years of age, his extraordinary talents were accidentally discovered by the duke, who found him in
the garden reading Newton's Principia in the Latin language ; when, on inquiry, the duke learned
that by procuring books, he had made himself master of arithmetic, geometry, &c. as well as the
Latin and French languages. Delighted with his conduct and conversation, tlie duke drew him from
obscurity, and placed him in a situation to pursue his favourite studies.

The time of Mr. Stone's birth is unknown, though it was probably towards the latter end of the
17th century, as the first edition of his Mathematical Dictionary was printed in 1726. After which,
several other useful works, both translations, and books of his own composition, follow at certain in-
tervals of time. As, 2. — A Treatise on Fluxions, in 1vol. 8vo. 1730; the first part being a trans-
lation, from the French, of I'Hopital's Analyse des Infiniments Petits; and the 2d part, or inverse
method, being supplied by Stone himself.

3. The Elements of Euclid, in 2 vols. 8vo. 1731 : being a neat and useful edition of those ele»
ments, with an account of the life and writings of Euclid, and a defence of his elements against mo-
dern objectors.

4. Dr. Barrow's Geometrical Lectures, translated from the Latin, in 1 vol. 8vo. 1735. Besides
several other smaller works.

The time of his death is also unknown.



VOL. XH.j PHILOSOPHICAL TRANSACTIONS. 393

xyy = bx- + ex -{• d, will, as they say, give 2 liyperbolo-parabolical figures,
equally distant on each side the diameter a b. See the 57th figure in Newton's
Treatise, and this is his 53d species, and Stirling's 57th.

1. If the equation bx^ — cj; + (/= O, have 2 equal roots, both with the sign
-j- ; the equation xyy ^ boe^ — ex + d, will, as they say, give 1 hyperbolo-para-
bolical curves, crossing each other at the point t in the diameter. See fig. 58
in Newton ; and this is his 45th species, and Stirling's 58th.

3. But if the equation bx"^ -\- ex -\- d=. o have 1 possible unequal negative
roots Ap and At, the curve given by the equation xyy = + bx'^ -{- ex -\- d, will
consist of 2 hyperbolo-parabolical parts, as also of an oval on the contrary side
the asymptote or principal absciss, as fig. 1, pi. 9. And this is one of the
species omitted by Sir Isaac and Mr. Stirling, which is really the 59th
species.

4. Also if the equation bx' -^ ex -^ d := O, have two equal negative roots Ap
and At; the curve given by the equation xi/y := + bx^ -\- ex + d, will consist of
2 hyperbolo-parabolical parts, and also of a conjugate point on the contrary
side the asymptote or principal ordinate, as figS2. And this is the other spe-
cies of these curves omitted by Sir Isaac and Mr. Stirling, which is really the
60th species.

5. If the roots of the equation bx'^ — ex -\- d = O, be real, and unequal, hav-
ing both the sign -f-; the curve given by the equation xyy = bx"^ — ex -{- d, will,
as they say, consist of a conchoidaThyperbola and a parabola, on the same side
the asymptote or principal ordinate. See fig. 59 in Newton ; and this is really
the 6 1st species.

6. If the roots of the equation bx^ + cx — d= O, have contrary signs, the
equation xyy = bar' + ex — d, will, as they say, give a conchoidal hyperbola
with a parabola on the contrary side the asymptote or principal ordinate. See
fig. 60 in Newton ; and this is really the 62d species.

^n Acemint, by Mr. Harris, of several Alterations and Contrivances about the
Terrestrial Globe, to render it, as he thought, more commodious and useful in
Practice. N° 456, p. 321.

A New Method of improving and perfecting Catodioptrical Telescopes, by form-
ing the Speculums of Glass, instead of Metal. By Caleb Smith. N*" 456,
p. 326.

The imperfections of telescopes are attributed to two causes ; viz. the unfit-
ness of the spherical figure to which the glasses are usually ground, and the
different refrangibility of the rays of light.

VOL. VIII. 3 E



304 PHILOSOPHICAL Tl!AN3ACTIONS. [aNNO 1740.

The first of these defects only, was known to the writers on dioptrics, before
Sir Isaac Newton ; for which reason, as he informs us. Opt. Lect. 1 , 2, " they
imagined, that optical instruments might be brought to any degree of perfec-
tion, provided they were able to communicate to the glasses, in grinding, what
geometrical figure they pleased ; to which purpose various mechanical con-
trivances were thought of, whereby glasses might be ground into hyperbolical,
or even parabolical, figures ; yet nobody succeeded in the exact description of
such figures ; and had their success been answerable to their wishes, yet their
labour would have been lost (continues this incomparable mathematician) ; for
the perfection of telescopes is limited, not so much for want of glasses truly
figured, according to the prescriptions of optic authors, (which all men have
hitherto imagined) as because that light itself is an heterogeneous mixture of
dift^erently refrangible rays ; so that were a glass so exactly figured as to collect
any one sort of rays into one point, it could not collect those also into the
same point, which having the same incidence upon the same medium, are apt
to sufi^er a difl^erent refraction." Phil. Trans. N° 80. And again, — " Diversa
diversorum radiorum refrangibilitas impedimento est, quo minus optica, per
' figuras, vel sphaericas, vel alias, perfici possint ; nisi corrigi possint errores
illinc oriundi, labor omnis in caeteris corrigendis imperite collocabitur," Prin-
cipia, &c. Scholium ad finem Libri Primi.

Now, for this principal and last-mentioned defect, no one has proposed any
remedy; apprehending perhaps the difficulty of attaining such to be insuperable ;
inasmuch as the great author of this discovery himself had not shown any me-
thod to correct those errors which arise from this inequality of refraction ; but
rather discouraged any such attempts, by declaring, " that on this account he
laid aside his glass-works," (Phil. Trans, N° 80) " and looked upon the im-
provement of telescopes, of given lengths, by refraction, as desperate." Optics,
ad Edit. p. 91.

However, as it has been proved by incontestible experiments, that this dis-
sipation of the rays of light, from whatever cause it proceeds, in passing out of
one medium into another, is not accidental and irregular ; but that every sort of
homogeneal rays, whether more or less refrangible, considered apart, are re-
fracted according to some constant uniform and certain law ; and as the re-
moval of so great an impediment as this, of unequal refraction in the rays of
light, is of great importance to the science of dioptrics, and absolutely neces-
sary to its further advancement ; we have thought it worthy of a careful exami-
nation, whether, in some cases at least, it might not be possible for contrary
refractions so to correct each other's inequalities, as to make their difference
regular ; and if this could be conveniently eflTected, Sir Isaac Newton has
acknowledged, " there would be no further difficulty." Phil. Trans. N* 88.



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 3^5

Now, on a clue consideration of this subject, we have found it possible, by
proper methods and expedients, to rectify those errors which proceed from the
different degrees of refrangibility in different rays, passing from one medium
into another ; admitting only this well-known and established principle, on
which we ground our reasoning, viz. " That the sines of refraction of rays,
differently refrangible, are one to another in a given proportion, when their
sines of incidence are equal." Optics, 2d edit. p. 66. And our present design
is, to show what advantage this will yield towards improving and perfecting ca-
todioptrical telescopes, by making the speculums of glass, instead of metal, in
the following manner: let a b c D b f, fig. 3, pi. Q, represent the section of a
concavo-convex speculum, whose two surfaces are segments of unequal spheres;
call the radius of the sphere, to which the concave side is ground, a ; and the
radius of the convex surface, which must be quicksilvered over, e ; let b r be
the axis of the speculum, or a line perpendicular to both the surfaces ; where
let p be the principal focus, or point where parallel rays of the most refrangible
kind are collected, by this speculum ; and a the focus, or point of concourse,
of such rays as are least refrangible ; viz. after they have suffered two refrac-
tions, at entering into, and passing out of, the concave surface d e p, and also
one reflection from the convex surface a b c. If the radius of concavity be
greater than the radius of convexity, as we will in the first place suppose, then
p will fall nearer the vertex of the speculum than the point a ; and the interval
a p will be the greatest aberration, or error, occasioned by the separation, or
unequal refraction, of the greatest and least refrangible rays, after their emer-
gence from the concave surface fed. Call the common sine of incidence, n ;
the sine of refraction of the least refrangible rays, out of a dense medium into a
rarer, m ; and of the most refrangible, n* ; then, according to the known and
received laws of refraction and reflection, the focal distance of the most refran-
gible rays, from the vertex of the speculum, neglecting its thickness, as of little
or no moment in the present case, will be found ^ r = p b. And

•^ ' {a—e) 2/«,-(-2«e

the quantity of the greatest aberration, occasioned by the difl^erent refrangibility
of the most and least refrangible rays, p a, will be to the focal distance just
mentioned, ? b, as (« — e) x(/* — m) to (a — e) m-{-en ; which quantity, or error,
thus obtained, to abbreviate the calculation, call i ; and now let it be required
to form a lens, if possible, which, placed at some given point in the axis, be-
tween the focus of the most refrangible rays p, and the vertex of the speculum,
as H, shall refract not only the rays of the most refrangible kind tending to the
point p, but also the rays of the least refrangible kind tending to a, in such a
manner, that both sorts shall concur, after such refraction, in some other point

3 £ 2 ■'-■ '-' 1^ - ■ ' ■



396 PHILOSOPHICAL TRANSACTIONS. [aNNO 1740.

of the axis r : let h p, the given distance of the point in the axis h, from the
focal point p, be called d; and then if the point h has been assumed, so that

the said given quantity, or distance, d, is greater than ~^ , but less than
-^^, the refracting superficies g h i, that shall perform what was required, will
be part of a concave sphere, whose radius is =.— ~ l"-— " . ^^^ jjg ^^g jjg.

r ' ' m(—(f/, — m)d '

tance of the given point h, from r, the point to which all the rays will tend,
after refraction at the said concave surface, (whose radius being found, as above,
we call v) will be = ,..1^"_ ■ • Lastly, upon the point r thus obtained,
as a centre, with an interval a little less than h b, describe the circumference
K L M, and the figure g h i m l k will denote the section of a double concave
lens, which, placed at the given point in the axis h, (taken nevertheless within
the limits above mentioned) will collect all sorts of rays proceeding from the
speculum, into one and the same focus, or point of the axis r, as was required;
for the surface g h i, which first receives those rays, will refract the most re-
frangible sort converging to the point p, and also the least refrangible converg-
ing towards q, so that both sorts, after such refraction, will concur in the point
B ; but the rays tending to r, it is manifest, will suffer no refraction at their
emergence from the superficies k l M, because r is the centre thereof, by con-
struction ; which point r, where a perfect image of an object infinitely dis-
tant will be formed, we call the focus of the telescope, to distinguish it from
the point p, which we have before called the focus of the speculum.

In this manner a lens, (or instead thereof a triangular prism with two of its
sides ground concave, and the third plain, if that be found as practicable) may
be formed and situated, so as to correct the errors of the speculum arising from
the different refrangibility of the rays of light. But in order to render this kind
of telescopes absolutely perfect in their construction, the errors also that result
from the spherical figure, must be rectified ; and with regard to this, we assert,
that it is possible to assume a point in the axis, between the focus of the spe-
culum and its vertex, (as we have taken the point h, in the following ex-
ample, see fig. 4,) at which, if a refracting superficies, or lens, be constituted,
according to the method already delivered, it will not only correct the errors
occasioned by the unequal refraction of the rays of light, but also rectify such
as proceed from the spherical figure of this speculum, to a much greater degree
of exactness than is requisite for any physical purpose, meaning always the
errors of those rays which respect the axis. Now to find or determine this
point, aflxjrds a problem not easy to be solved ; and we recommend it, as worthy
of the consideration of geometricians.



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 8p7

Seeing therefore it is possible, and we believe also practicable, to remedy the
imperfections of this kind of speculums, from whatsoever cause they arise, by
the method we have here proposed ; it seems to follow, that catodioptrical te-
lescopes may be carried, by this means, to as great a degree of perfection, as
they are capable of receiving ; provided spherical figures can be truly given,
with an exquisite polish, to glasses of a large aperture, and a foil of quicksilver
made also to retain that figure accurately, and without any inequality ; for the
object glass or speculum being rendered perfect, so as that all sorts of rays,
proceeding from one lucid point in its axis, shall be collected by means of the
lens exactly in another point, its aperture may then be extended to its furthest
limits ; and that is, till the whole pupil of the eye, or the whole portion of
the eye-glass to be used, when that becomes necessarily less than the pupil,
be filled with rays proceeding from the speculum, and flowing from one point
of the object, but no farther; because this is a limitation made by nature in
the structure of the eye itself : and in telescopes whose construction is such as
we have now described, the largest aperture of the speculum that can ever be
of use, will be to the diameter of the pupil of the eye, very nearly, in a ratio
compounded of the ratios of the focal length of the speculum, to the distance
of that focus from the lens, and of the distance of the lens from the focus of
the telescope, to unity : that is, of bp to ph, and of kh to ) ; which proportion
holds, whatever be the charge or the power of magnifying.

But if inquiry be made as to the charge most proper and convenient, that
will be best determined by experience, in these, as well as in all other sorts of
telescopes : however, on supposition that one of a given length has its aperture
and charge rightly ordered and proportioned, the rule for preserving the same
degree of brightness and distinctness, in all others of a like construction, will
be, to make the apertures, and magnifying powers, directly as the focal lengths
of the speculums; which shows the vast advantage and perfection of these te-
lescopes, above the common reflecting ones; where, according to Sir Isaac
Newton's rule, the apertures, and powers of magnifying, must be as the bi-
quadrate roots of the cubes of their lengths. See his Optics, 2d edition,

P- 97-

It is likewise a considerable advantage in this construction, that the reflection
from the concave side of the speculum will do no sensible prejudice; because
the image of any object there made, is removed to so vast a distance from the
principal image, formed by the convex surface, as to create no manner of con-
fusion or disturbance in the vision ; which necessarily happens, in some degree,
from the vicinity of those images, when the glass is ground concave on one



3^ PHILOSOPHICAL TKANSACTIONS, [aNNO J740.

side, and as much convex on the other ; according to the method propounded
by Sir Isaac Newton, in his most excellent book, of Optics.

It may be imagined perhaps, at first view, that, if our reasoning is just, the
errors of refracting telescopes, occasioned by the different refrangibility of
light, may be corrected by a like artifice : but the aberration of the rays from
the principal focus is there so great, and bears so considerable a proportion to
the focal length of the telescope, that the error cannot be rectified by the in-
terposition of any lens, until the rays are, by a contrary refraction, collected
again at an infinite distance, which renders this expedient quite useless: how-
ever, there is no need to despair of accomplishing even this, by other methods :
and, by the way, we may observe, if it were worth while to seek a remedy for
the errors occasioned by the spherical figure of the object-glass only, in diop-
trical telescopes ; that might be obtained by the proper application of a suitable
lens, between the focus and the vertex of the object-glass ; which is much
more easy and practicable, than the grinding of glasses to hyperbolical or ellip-
tical figures.

For a further illustration of the foregoing, it may be proper to exhibit the
several parts and proportions of a telescope in numbers, computed according to
the theorems already delivered ; and in practice we judge it will be most con-
venient, that the radii of the spheres, to which the concave and convex sides
of the speculum are ground, be nearly in the ratio of 6 to 5 ; as in the follow-
ing example ; where, in fig. 4,



Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 46 of 85)