ART. V. Philosophical Transactions of the Royal Society of London, for the Year 18co. Part I. 4to. sewed. Elmsly and Bremner.

A

CCORDING to our former practice, we shall divide the papers in this volume into classes; and we shall now begin with those which range under the head of

MATHEMATICS.

On the Method of determining, from the real Probabilities of Life, the Values of Contingent Reversions in which three Lives are involved in the Survivorship. By William Morgan, Esq. F. R.S.

In this paper, the ingenious author pursues his investigation of the doctrine of contingent reversions, as they are applicable to those cases in which three lives are concerned in the survivorship; and the problems now presented to the public, added to those contained in his former papers, afford a complete view of the subject to persons who are conversant in this department of science. As Mr. Morgan's eminence in disquisitions of this nature is universally acknowleged, it may be sufficient to observe, on the present occasion, that we are indebted to him for the first solutions that have ever been deduced, in` the case of two and three lives, from just principles, and the real probabilities of life; and that, with respect to many of the problems, none have ever attempted so much as to approximate to the value of the reversion. 'Being possessed (says Mr. M.) of correct solutions of all the cases in which two or three lives are involved in the survivorship, we are possessed of all that is really useful, and therefore I feel the greater satisfaction in closing my inquiries on this subject. For, in regard to contingencies depending on four or more lives, the cases are not only much too numerous and intricate to admit of a solution, but they occur so seldom in practice, as to render the entire investigation of them, were it even possible, a matter of little or no importance.'

Of Mr. Hellins's second Appendix to the improved Solution of a Problem in Physical Astronomy, inserted in the Philosophical Transactions for the Year 1798, we shall only say that it is a valuable addition to his former communications on the same subject; as it furnishes improved formula for facilitating and abridging the calculations that are necessary in the solution of the problem to which they relate. Sir Godfrey Copley's medal has been adjudged to Mr. H. for this solution, and his other mathematical papers.

PHILOSOPHY.

On the Power of penetrating into Space by Telescopes; with a. comparative Determination of the Extent of that Power in Natural

Vision,

Vision, and in Telescopes of various Sizes and Constructions; illustrated by select Observations. By William Herschel, LL.D.

F.R.S.

In order to proceed with clearness and facility in the discussion of the subjects of this paper, the author begins with defining the terms which most frequently occur, and explaining some algebraic symbols which he has adopted for expressing them. He ascribes brightness to bodies that throw out light, and those which throw out most light are the brightest. The whole quantity of light thrown out by a luminous surface is called L; and the luminous physical points which compose this surface are denoted by N. If the copiousness of the emission of light from each of these points were precisely the same, it might be expressed by C: but, as that is most probably never the case, C is made to signify the mean quantity, Hence it appears that CN is equal to L; and brightness will consequently be truly defined by CN. In estimating the appearance of luminous objects at any assigned distance, it will be proper to leave out of the account every part of CN, which is not applied to the purpose of vision; and, therefore, as L represents the whole quantity of light thrown out by CN, that part of it which is used in vision, either by the eye or by a telescope, is denoted by . Accordingly, the equation of light, in this sense of it, is CN=1. As the density of light, however, decreases in the ratio of the squares of the distances of the luminous objects, the quantity of it at the distance D will be expressed by Dz

In natural vision, the quantity / undergoes a considerable change by the opening and contracting of the pupil of the eye. If we call the aperture of the iris a, it is known to vary considerably in different persons. The variation of a in different circumstances is not easily ascertained: but, in determining the quantity of light admitted through a telescope, no such difficulty occurs. This must depend on the diameter of the object-glass, or mirror; and its aperture A may at all times be duly measured. Hence it follows that the expression will always be accurate for the quantity of light ad

a'l

D

After having answered some objections to the theory of brightness hère proposed, the author proceeds to investigate

the general extent of natural vision.

REV. Nov. 1800.

Among

;

Among the reflecting luminous objects, our penetrating powers are sufficiently ascertained. From the Moon we may step to Venus, to Mercury, to Mars, to Jupiter, to Saturn, and last of all to the Georgian planet. An object seen, by reflected light at a greater distance than this, it has never been allowed us to perceive: and indeed it is much to be admired, that we should see borrowed illumination to the distance of more than 18 hundred millions of miles especially when that light, in coming from the Sun to the planet, has to pass through an equal space before it can be reflected, whereby it must be so enfeebled as to be above 368 times less intense on that planet than it is with us, and when probably not more than one-third part of that light can be thrown back from its disk. For, according to Mr. Bouguer, the surface of the Moon absorbs about two-thirds of the light it receives from the Sun.

. The range of natural vision with self-luminous objects, is incomparably more extended, but less accurately to be ascertained. From our brightest luminary, the Sun, we pass immediately to very distant objects; for Sirius, Arcturus, and the rest of the stars of the first magnitude, are probably those that come next; and what their distance may be, it is well known, can only be calculated imperfectly from the doctrine of parallaxes, which places the nearest of them at least 412,530 times farther from us than the Sun.'

Next to these stars of the first magnitude, a second set will occur; of which the situation, one with another, may betaken at about double the distance of the former from us. If we suppose a Cygni, ß Tauri, &c. to belong to this class of the second magnitude, it is known that, whether we look at these or at the former, the aperture of the iris will probably undergo no change; and therefore a becomes a given quantity, and may be left out in the above expression for the bright

a2l

D2

ness of these objects. D, according to the preceding statement of distance, will be in one case four times that of the other; and consequently the two expressions for the brightness of the stars will be for those of the first magnitude, and for those of the second.

The quantities being thus prepared, what I mean to suggest (says the author) by an experiment is, that since sensations, by their nature, will not admit of being halved or quartered, we come thus to know by inspection, what phenomenon will be produced by the fourth part of the light of a star of the first magnitude. In this sense, I think we must take it for granted, that a certain idea of brightness, attached to the stars which are generally denominated to be of the second magnitude, may be added to our experimental knowledge; for, by this means, we are informed what we are to a2l al understand by the expressions

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Sirius

a'l

Taurit

The names of the objects, Sirius, & Tauri, are here used to

express their distance from us.'

We

We cannot wonder at the immense difference between the brightness of the Sun and that of Sirius; since the two first expressions, when properly resolved, give us a ratio of brightness of more than 170 thousand millions to one; whereas the two latter, as has been shewn, give only a ratio of four to one.'

The author then proceeds to ascertain the comparative brightness of stars of the third magnitude, such as the pole-star, y Cygni, Bootis, and others of the same order; and he finds that the difference between these and the stars of the preceding order is much less striking than that between the stars of the first a'l and second magnitude;' and that the expressions

B'Tauri

are not in the high ratio of 4 to 1, but only

as 9 to 4 or 2 to 1.'

From the faintness of stars of the 7th magnitude, Dr. H. infers that no star, 8, 9, or at most 10 times as far from us as Sirius, can possibly be perceived by the natural eye :-but, when the light of single stars fails, the united lustre of sidereal, systems will still be perceived. Such are the whitish patch. in the sword-handle of Perseus; beyond this the cluster discovered by M. Messier in 1764, north following H Geminorum; at a still farther distance, the nebula between ท and C Hercules, discovered by Dr. Halley in 1714; and among the farthest objects that can make an impression on the eye,' unaided by telescopes, the nebula in the girdle of Andromeda, discovered by Simon Marius in 1612.

Dr. Herschel's next object is to ascertain the penetrating power of the telescopes. In a telescope, the brightness or light admitted, which to the naked eye is truly represented by

al

D

A l
D-

will be expressed by and therefore the artificial power of penetrating into space should be to the natural one as A to a: but this proportion must be corrected by the prac tical deficiency in light reflected by mirrors, or transmitted through glasses. Dr. H. investigates what allowances are to be made on this account; and he then deduces a general formula, which expresses the penetrating power of all sorts of telescopes compared with that of the natural eye, as a standard, according to any supposed aperture of the iris, and proportion of light returned by reflection, or transmitted by refraction. He proceeds to determine the powers of the instruments used by himself in his astronomical observations; and for this pur pose he has reported the observations themselves, arranged in a manner which seemed to be the best adapted to his purpose.

He has also established the distinction subsisting between magnifying power and the power of penetrating into space, and has evinced the difference between them in a variety of instances. He shews how these powers occasionally interfere with each other, so that in some instances the magnifying power is injured by the penetrating power, and vice versas and he observes that the highest power of magnifying maypossibly not exceed the reach of a 20 or 25 feet telescope, or may lie in even a less compass, allowing for some favourable hours, in which it is hardly possible to set a limit to magnifying power. With respect to the penetrating power of telescopes, he is of opinion that there is room for a considerable increase. The penetrating power of his 40 feet reflector already extends to 191,69, and he thinks that it might be carried to 500, but probably not much farther. The natural limit seems to be an equation between the faintest star that can be made visible by any means, and the united brilliancy of star-light.' He concludes, on the whole, that objects are viewed in their greatest perfection, when, in penetrating space, the magnifying power is so low as only to be sufficient to shew the object well; and when in magnifying objects, by way of examining them minutely, the space-penetrating power is no higher than what will suffice for the purpose; for in the use of either power, the injudicious overcharge of the other will prove hurtful to perfect vision."

The author terminates this paper with a calculation of the time which it would require to sweep the heavens as far as they are within the reach of his 40 feet telescope, charged with a magnifying power of 1000. Under all the circumstances that occur, a year which will afford yo, or at most 100 hours, may be deemed very productive; and with this allowance, it appears that it will require not less than 598 years to look with the 40 feet reflector, charged with the above-mentioned power, only one single moment into each part of space; and even then, so much of the southern hemisphere will remain unexplored, as will take up 213 years more to examine.'

Outlines of Experiments and Inquiries respecting Sound and Light. By Thomas Young, M. D. F.R.S.

The first section of this paper contains an estimate of the quantity of air discharged through an aperture, which the author determines to be nearly in the sub-duplicate ratio of the pressure; and he ascertains the ratio of the expenditures by different apertures, with the same pressure, to lie between the ratio of their diameters and that of their areas. The apparatus