D2:

4D2

D1 + 2D2 (2c2 — p2 — 82) — — p1 — 84 + Qμ·282,

=

— (2c2 — r2 —♪2)±√4c1 +r1 +d2 — 4 c2 r2 — 4c2 82 + 48a‚2, D2=— (c+r) (c—r) — (c—d) (c+d) ±2√c1 —c2 r2 —c2 d2+d2r2, c1 — c2 μ‚2 — c2x2 + d12 μ2 = c2 (c2 — d2) — 7·2 (c2 — d'2) =

(c2 — r2) (c2 — d12)=(r+c) (r−c) (8 + c) (8 —c), D2=(r+c)(r−c)+(d+c) (d—c)±√(r+c)(r−c)(d+c)(d—c),

(d+c) (d—c)

A'=(r+c) (r—c) B'= (8+ c) (8-c)

D2=A'+B'+2√A'B' D2 = a2 + ß2 = (4,2 + x,2)ť2 + 2(^4, + xx,)t + A2 + x2,

whence, (making — ▲▲, —λa, — A_42+λ2=B)

A

The time of the middle of the eclipse is = T +

B

24

The duration of the eclipse is = 2√ (1) *

B

(c being

2

+

B

The shortest distance between the centre of the Moon and of

the section of the conus umbræ (occurring when t is equal to

the time of the middle of the eclipse, or t=T+4),

given by the value of D, derived from the equation

will be

Lastly, any enlightened quantity of the Moon, or any distance of the horns, will be given by the formula

observing, that, in the former case, D = r —d' + ε, in the latter, D2 = A' + B' + 2√A'B',

A' (r+c) (rc) B'=(+c) (d' — c) ; thus we may have the time t expressed in function of, or in function of c.

The longitude of the place will be expressed in time by the formula Lt-t', in which t' represents the time of the observation of or c; the longitude being east or west, according as L is positive or negative.

Finally, substituting the value of t, we have the longitude expressed in time by the formula,

*

I have calculated M. de Humboldt's observation at Ibague by this formula, and the result would no doubt accord precisely with that in the text, were the elements it contains free from errors; for, after correcting the most palpable, my result differs but 27" from that of M. Oltmans. The following errors cannot be disputed, and other lesser ones certainly exist.

It is impossible that 21h 20′ 45′′ can be the mean time at Paris, since the elements are calculated very near the opposition, and this happened, according to the text, at 19h 26′ 41′′. If we suppose, for a moment, that this latter element is inexact, we may still convince ourselves that 21h 20′ 45′′ cannot be the correct time, by merely observing, that the enlightened part of the

Voyage de Humboldt, Partie Astronomique, (2 vol. 4to), vol. ii. p. 255.

Moon 23′ 30′′ must be near the end of the eclipse, since both the time and the quantity of the enlightened part of the Moon continue augmenting; nevertheless the time of this measurement is 21h 0′ 13′′.9, i. e. less than 21h 20′ 45′′, which is evidently absurd. This error of the text leaves no other alternative than to deduct the time for which the observation is calculated from that of the full Moon, which gives us 19h 27′ 28′′.8. evident error of the text is detected as follows:

Another

21h 0' 13.9 according to the text,

15 50 54.9

ART. VIII.-Remarks on the Increase of the Population of the United States and Territories of North America, with Original Tables, deduced from the American Population Returns, to illustrate the various Rates of Increase in the White Population and Slaves, and also the comparative degrees in which Agriculture, Commerce, and Manufactures prevail. By GEORGE HARVEY, Esq. M. G. S. M. A. S., &c. (Continued from Vol. VIII. p. 339.)

SLAVE POPULATION.

EVERY lover of humanity necessarily feels an interest in the

condition of that class of our fellow men who are doomed to spend the long range of a miserable existence in a state of slavery. The heart sympathises with their misfortunes, and we eagerly embrace every opportunity, which is likely to throw even but a feeble and uncertain light on their unhappy condition. The returns of the slaves contained in the American Population Tables, are probably the only sources from which any satisfactory information can be drawn, to illustrate this very important subject.

It has been already remarked, that in the enumerations of the American people, prior to the census of 1820, the slaves were thrown into one mass, without any distinction as to their sexes,

1

and much less any divisions relating to their ages. In the census of 1820, however, these very important particulars were attended to; the males being separated from the females, and each sex divided into the four classes alluded to at the commencement of the paper. It was observed also, at the same time, as a circumstance much to be regretted, that the ages of the slaves do not entirely correspond with the classes into which the free males and females were divided; and that, therefore, many interesting comparisons which might otherwise have been made, with the white part of the population, could not, under the present circumstances, be instituted. It would have been interesting, for example, to have been enabled to compare the slave population under ten years of age, with the free white population of the same class, and by this means to have considered, how far the general circumstances of the younger classes of slaves at all assimilate to the state of the free American population of corresponding ages; whether slavery exerts any influence of a very baneful nature on the young; and whether the freedom of the parent does not contribute, in a very powerful degree, to cherish a healthy and vigorous offspring. Some idea of the nature of this influence may be drawn from a Table which will speedily follow. Two classes, however, of the slave population, admit of a direct comparison with the free population; that is, those of twenty-six and under forty-five, and of forty-five and upwards. If, indeed, we regard the aggregate of the classes below the former of these, in each case, as a single class, another comparison may be made, with each sex, below the age of twenty-six.

If we take the total amount of each class of the free. white population, in all the provinces, in 1820, it appears, that, for every 100 free white males of 26 and under 45, there were 354 under 26 years of age, and 65 of 45 and upwards; but that for every 100 male slaves of 26 and under 45, there were 334 under 26 years of age, and only 47 of forty-five years and upwards. So also, for every 100 free females of 26 and under 45, there were 362 under 26, but only 63 of 45 years and upwards; and for every 100 female slaves of 26 and under 45, there were 345 under 26 years of age, and 46 of 45 years and upwards.

These interesting results will be more clearly perceived in the following Table:

These numerical results clearly prove the baneful effects of hard labour and coercion on the unfortunate slaves; and is most manifest in the class of 45 and upwards, the representative numbers for the male and female slaves being so very much below the corresponding numbers for the free population. If we admit, however, for a moment, that the relation of the representative numbers of the male and female slaves, to the corresponding numbers of the free white population, be such as the laws of nature allow; and that slavery exerts no improper influence on the condition of man, then ought the representative numbers belonging to the classes under 26, and of 45 and upwards, in the free persons and slaves, to bear some analogy to each other. This, unfortunately for the interests of humanity, is not the case; for if we take the males of the classes here alluded to, we shall find, that if the slaves of 45 and upwards bore the same relation to the free males of that age, as the slaves under 26 do to the free males of the same class; then, instead of there being only 47 for the representative number of the oldest class of slaves, there should have been 61; and if a similar comparison be made for the females, we shall find, that, instead of the representative number for the oldest class being 45, it should have been 60. Hence it appears there is a deficiency of no less than 14 persons of each sex, in relation to the assumed number for the class of 26 and under 45, occasioned, it may be fairly said, by the hard labour, and the many miseries necessarily attendant on slavery. This is a point of view, however, much too favourable, for, considering the subject, no other reason can be assigned, why the representative numbers for the slaves under 26 are also below the corresponding numbers for the free persons, but that the effects of slavery have made them VOL. IX. NO. 17. JULY 1823.

E