Chapter IX: Part 9

Make, now, _p_³ = 1, _q_³ = 1, and _r_³ = 1; that is, let _p_, _q_, and _r_, be the three roots of the cubic equation _z_³ = 1, or _z_³ - 1 = 0: then, seeing both the second and third terms of this equation are wanting, not only the sum of all the roots (_p_ + _q_ + _r_) but the sum of all their squares (_p_² + _q_² + _r_²) will vanish, or be equal to nothing (by common algebra), as they ought, to fulfil the conditions of the two first equations. Moreover, since _p_³ = 1, _q_³ = 1, and _r_³ = 1, it is also evident, that _p_⁴ + _q_⁴ + _r_⁴ (= _p_ + _q_ + _r_) = 0, _p_⁵ + _q_⁵ + _r_⁵ (= _p_² +_q_² + _r_²) = 0, _p_⁶ + _q_⁶ + _r_⁶ (= _p_³ + _q_³ + _r_³) = 3. Which equations being, in effect, nothing more than the first three repeated, the values of _p_, _q_, _r_, above assigned, equally fulfil the conditions of these also: so that the series arising from the addition of three assumed ones will agree, in every term, with _that_ whose sum is required: but those series’ (whereof the quantity in question is composed) having all of them the _same form_ and the _same coefficients_ with the original series _a_ + _bx_ + _cx_² + _dx_³, &c. (= _S_), their sums will therefore be truly obtained, by substituting _px_, _qx_, and _rx_, successively, for _x_, in the given value of _S_. And, by the very same reasoning, and the process above laid down, it is evident, that, if every _nᵗʰ_ term (instead of every third term) of the given series be taken, the values of _p_, _q_, _r_, _s_, &c. will then be the roots of the equation _zⁿ_ - 1 = 0[155]; and that, the sum of all the terms so taken, will be truly obtained by substituting _px_, _qx_, _rx_, _sx_, &c. successively for _x_, in the given value of _S_, and then dividing the sum of all the quantities thence arising by the given number _n_.

The same method of solution holds equally, when, in taking every _n_ᵗʰ term of the series, the operation begins at some term after the first. For all the terms preceding _that_ may be transposed, and the whole equation divided by the power of _x_ in the first of the remaining terms; and then the sum of every _nᵗʰ_ term (beginning at the first) will be found by the preceding directions; which sum, multiplied by the power of _x_ that before divided, will evidently give the true value required to be determined. Thus, for example, let it be required to find the sum of every third term of the given series _a_ + _bx_ + _cx_² + _dx_³ + _ex_⁴, &c. (= _S_), beginning with _cx_². Then, by transposing the two first terms, and dividing the whole by _x_², we shall have _c_ + _dx_ + _ex_² + _fx_³, &c. = (_S_ - _a_ - _bx_) ⁄ (_xx_) (= _S´_). From whence having found the sum of every third term of the series _c_ + _dx_ + _ex_² + _fx_³, &c. beginning at the first (_c_), that sum, multiplied by _x_², will manifestly give the true value sought in the present case.

And here it may be worth while to observe, that all the terms preceding _that_ at which the operation (in any case) begins, may (provided they exceed not in number the given interval _n_) be intirely disregarded, as having no effect at all in the result. For if in that part ((-_a_ - _bx_) ⁄ _xx_) of the value of _S´_, above exhibited, in which the first terms, _a_ and _bx_, enter, there be substituted _px_, _qx_, _rx_, successively, for _x_ (according to the _prescript_) the sum of the quantities thence arising will be

- _a_ ⁄ (_p_²_x_²) - _a_ ⁄ (_q_²_x_²) - _a_ ⁄ (_r_²_x_²)
- _b_ ⁄ _px_ - _b_ ⁄ _qx_ - _b_ ⁄ _rx_

which, because _p_³ = 1, _q_³ = 1, &c. (or _p_² = 1 ⁄ _p_, _q_² = 1 ⁄ _q_, &c.) may be expressed thus;

- _a_ ⁄ _xx_ × (_p_ + _q_ + _r_)
- _b_ ⁄ _x_ × (_p_² + _q_² + _r_²)

But, that _p_ + _q_ + _r_ = 0, and _p_² + _q_² + _r_² = 0, hath been already shewn; whence the truth of the general observation is manifest. Hence it also appears, that the method of solution above delivered, is not only general, but includes this singular beauty and advantage, that in all series’ whatever, whereof the terms are to be taken according to the same assigned order, the quantities (_p_, _q_, _r_, &c.), whereby the resolution is performed, will remain invariably the same. The greater part of these quantities are indeed _imaginary_ ones; and so likewise will the quantities be that result from them, when substitution is made in the given expression for the value of _S_. But by adding, as is usual in like cases, every two corresponding values, so resulting together, all marks of _impossibility_ will disappear.

If, in the series to be summed, the alternate terms (_viz._ the 2d, 4th, 6th, _&c._) should be required to be taken under signs contrary to what they have in the original series given; the reasoning and result will be no-ways different; only, instead of making _p_³ + _q_³ + _r_³ (or _pⁿ_ + _qⁿ_ + _rⁿ_, &c.) = +3 (or +_n_), the same quantity must, here, be made = -3 (or -_n_). From whence, _pⁿ_ being = -1, _qⁿ_ = -1, &c. the values of _p_, _q_, _r_, &c. will, in this case, be the roots of the equation _zⁿ_ + 1 = 0.

It may be proper, now, to put down an example, or two, of the use and application of the general conclusions above derived. First, then, supposing the series, whose sum is given, to be _x_ + _x²_ ⁄ 2 + _x³_ ⁄ 3 + _x⁴_ ⁄ 4 ... + _xᵐ_ ⁄ _m_ + _xᵐ ⁺ ¹_) ⁄ (_m_ + 1) + _xᵐ ⁺ ²_ ⁄ (_m_ + 2) ... + _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ⁿ ⁺ ¹_ ⁄ (_m_ + _n_ + 1) +, &c. = - H. Log.(1-_x_) (= _S_); let it be required, from hence, to find the sum of the series (_xᵐ_ ⁄ _m_ + _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) &c.) arising by taking every _nᵗʰ_ term thereof, beginning with that whose exponent (_m_) is any integer less than _n_. Here, the terms preceding _xᵐ_ ⁄ _m_ being transposed, and the whole equation divided by _xᵐ_, we shall have 1 ⁄ _m_ + _x_ ⁄ (_m_ + 1) + _x_² ⁄ (_m_ + 2) + _x_³ ⁄ (_m_ + 3), &c. = -(1 ⁄ _xᵐ_) × H. Log.(1 - _x_) - (_x_ + ½_x_², &c.) ⁄ _xᵐ_. In which value, let _px_, _qx_, _rx_, &c. be, successively, substituted for _x_ (according to prescript) neglecting intirely the terms (_x_ + ½_x_²) ⁄ _xᵐ_, as having no effect at all in the result: from whence we get -1 ⁄ (_(px)ᵐ_) × Log.(1 - _px_) - (1 ⁄ _(qx)ᵐ_) × Log.(1 - _qx_) - (1 ⁄ _(rx)ᵐ_) × Log.(1 - _rx_), &c. Which multiplied by _xᵐ_ (the quantity that before divided) gives -1 ⁄ (_pᵐ_) × Log.(1 - _px_) - 1 ⁄ (_qᵐ_) × Log.(1 - _qx_) - 1 ⁄ (_rᵐ_) × Log.(1 - _rx_), &c. = _n_ times the quantity required to be determined.

But now, to get rid of the imaginary quantities _q_, _r_, &c. by means of their known values α + √(αα - 1), α - √(αα - 1), &c. it will be necessary to observe, that, as the product of any two corresponding ones (α + √(αα - 1) × (α - √(αα - 1)) is equal to unity, we may therefore write (α - √(αα - 1))_ᵐ_ (= _rᵐ_) instead of its equal 1 ⁄ (_qᵐ_), and (α + √(αα - 1))_ᵐ_ (= _qᵐ_) instead of its equal 1 ⁄ (_rᵐ_): by which means the two terms, wherein these two quantities enter, will stand thus; -(α - √(αα - 1))_ⁿ_ × Log. (1 - _qx_) - (α + √(αα - 1))_ᵐ_ × Log. (1 - _rx_).

But, if _A_ be assumed to express the co-sine of an arch (_Q_), _m_ times as great as that (360° ⁄ _n_) whose co-sine is here denoted by α; then will _A_ - √(_AA_ - 1) = [156](α - √(αα - 1))_ᵐ_, and _A_ + √(_AA_ - 1) = (α + √(αα - 1))_ᵐ_: which values being substituted above, we thence get

-_A_ × (log. (1 - _qx_) + log. (1 - _rx_)) + √(_AA_
- 1) × (log. (1 - _qx_) - log. (1 - _rx_));

whereof the former part (which, exclusive of the factor _A_, I shall hereafter denote by _M_) is manifestly equal to -_A_ × log. ((1 - _qx_) × (1 - _rx_)) (by the nature of logarithms) = -_A_ × log. 1 - (_q_ + _r_)._x_ + _qrx_² = -_A_ × log. (1 - 2α_x_ + _xx_) (by substituting the values of _q_ and _r_): which is now intirely free from imaginary quantities. But, in order to exterminate them out of the latter part also, put _y_ = log. (1 - _qx_) - log. (1 - _rx_); then will _ẏ_ = -_qẋ_ ⁄ (1 - _qx_) + _rẋ_ ⁄ (1 - _rx_) = -((_q_ - _r_) × _ẋ_) ⁄ (1 - (_q_ + _r_) × _x_ + _xx_) = -(2√(αα - 1) × _ẋ_) ⁄ (1 - 2α_x_ + _xx_) = -(2√(-1) × √(1 - αα) × _ẋ_) ⁄ (1 - 2αx + xx); where (√(1 - αα) × ẋ) ⁄ (1 - 2α_x_ + _xx_) expresseth the fluxion of a circular arch (_N_) whose radius is 1, and sine = (√(1 - αα) × _ẋ_) ⁄ (1 - 2α_x_ + _xx_); consequently _y_ will be = -2√(-1) × _N_: which, multiplied by √(_AA_ - 1), or its equal √(-1) × √(1 - _AA_), gives 2√(1 - _AA_) × _N_; and, this value being added to that of the former part (found above), and the whole being divided by _n_, we thence obtain (-_AM_ + 2√(1 - _AA_) × _N_) ⁄ _n_, or 1 ⁄_n_ × (-co-s. _Q_ × _M_ + sin. _Q_ × 2_N_) for that part of the value sought depending on the two terms affected with _q_ and _r_. From whence the sum of any other two corresponding terms will be had, by barely substituting one letter, or value, for another: So that,

{ -log. (1 - _x_)
{ -co-s. _Q_ × _M_ + sin. _Q_ × 2_N_
(1 ⁄ _n_) × { -co-s. _Q´_ × _M´_ + sin. _Q´_ × 2_N´_
{ -co-s. _Q´´_ × _M´´_ + sin. _Q´´_ × 2_N´´_
{ -&c. + &c.

will truly express the sum of the series proposed to be determined; _M_, _M´_, _M´´_ &c. being the hyperbolical logarithms of 1 - 2α_x_ + _xx_, 1 - 2β_x_ + _xx_, 1 - 2γ_x_ + _xx_, &c. _N_, _N´_, _N´´_ &c. the arcs whose sines are _x_√(1 - αα) ⁄ √(1 - 2α_x_ + _xx_), _x_√(1 - ββ) ⁄ √(1 - 2β_x_ + _xx_), _x_√(1 - γγ) ⁄ √(1 - 2γ_x_ + _xx_), &c. and _Q_, _Q´_, _Q´´_, &c. the measures of the angles expressed by (360° ⁄ _n_) × _m_, 2 × (360° ⁄ _n_) × _m_, 3 × (360° ⁄ _n_) × _m_, &c. And here it may not be amiss to take notice, that the series _xᵐ_ ⁄ _m_ + _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) + &c. thus determined, is that expressing the fluent of (_xᵐ ⁻ ¹ẋ_) ⁄ (1 - _xⁿ_); corresponding to one of the two famous _Cotesian forms_. From whence, and the reasoning above laid down, the fluent of the other _form_, _xᵐ ⁻ ¹ẋ_ ⁄ (1 + _xⁿ_), may be very readily deduced. For, since the series (_xᵐ_ ⁄ _m_ - _xᵐ ⁺ ⁿ_ ⁄ (_m_ + _n_) + _xᵐ ⁺ ²ⁿ_ ⁄ (_m_ + 2_n_) - _xᵐ ⁺ ³ⁿ_ ⁄ (_m_ + 3_n_) &c.) for this last fluent, is that which arises by changing the signs of the alternate terms of the former; the quantities _p_, _q_, _r_, &c. will here (agreeably to a preceding observation) be the roots of the equation _zⁿ_ + 1 = 0; and, consequently, α, β, γ, δ, &c. the co-sines of the arcs 180° ⁄ _n_, 3 × 180° ⁄ _n_, 5 × 180° ⁄ _n_, &c. (as appears by the foregoing note). So that, making _Q_, _Q´_, _Q´´_, &c. equal, here, to the measures of the angles (180° ⁄ _n_) × _m_, 3 × (180° ⁄ _n_) × _m_, 5 × (180° ⁄ _n_) × _m_, &c. the fluent sought will be expressed in the very same manner as in the preceding case; except that the first term, -log. (1 - _x_) (arising from the _rational_ root _p_ = 1) will here have no place.

After the same manner, with a small increase of trouble, the fluent of _xᵐ ⁻ ¹ẋ_ ⁄ (1 ± 2_lxⁿ_ + _x_²_ⁿ_) may be derived, _m_ and _n_ being any integers whatever. But I shall now put down one example, wherein the impossible quantities become exponents of the powers, in the terms where they are concerned.

The series here given is 1 - _x_ + _x_² ⁄ 2 + _x_³ ⁄ (2.3) + _x_⁴ ⁄ (2.3.4) - _x_⁵ ⁄ (2.3.4.5), &c. = the number whose hyp. log. is -_x_, and it is required to find the sum of every _nᵗʰ_ term thereof, beginning at the first. Here the quantity sought will (according to the general rule) be truly defined by the _nᵗʰ_ part of the sum of all the numbers whose respective logarithms are -_px_, -_qx_, -_rx_, &c.; which numbers, if _N_ be taken to denote the number whose hyp. log. = 1, will be truly expressed by _N⁻ᵖˣ_, _N⁻⒬ˣ_, _N⁻ʳˣ_, &c. From whence, by writing for _p_, _q_, _r_, &c. their equals 1, α + √(αα - 1), α - √(αα - 1), β + √(ββ - 1), β - √(ββ - 1), &c. and putting α´ = √(1 - αα), β´ = √(1 - ββ), &c. we shall have 1 ⁄ _n_ × (_N⁻ᵖˣ_ + _N⁻⒬ˣ_ + _N⁻ʳˣ_), &c. = 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × (_N⁻ᵃ‘ˣ_√⁻¹) + (_Nᵃ‘ˣ_√⁻¹) + _N⁻ᵝˣ_ × (_N⁻ᵝ‘ˣ√⁻¹_) + _Nᵝ‘ˣ√⁻¹_) + &c. But _N⁻ᵃ‘ˣ√⁻¹_ + _Nᵃ‘ˣ√⁻¹_ is known to express the double of the co-sine of the arch whose measure (to the radius 1) is α´_x_. Therefore we have 1 ⁄ _n_ into _N⁻ˣ_ + _N⁻ᵃˣ_ × 2 co-s. α´_x_ + _N⁻ᵝˣ_ × 2 co-s. β´_x_, &c. for the true sum, or value proposed to be determined.

The solution of this case, in a manner a little different, I have given some time since, in another place; where the principles of the general method, here extended and illustrated, are pointed out. I shall put an end to this paper with observing, that if, in the series given, the even powers of _x_, or any other terms whatever, be wanting, their places must be supplied with cyphers; which, in the order of numbering off, must be reckoned as real terms.

CIV. _Observatio Eclipsis Lunæ Die_ 30 Julii 1757. _habita Olissipone à_ Joanne Chevalier, _Congregationis Oratorii Presbytero, é Regia_ Londinensi _Societate. Communicated by_ Jacob de Castro Sarmiento, _M.D. F.R.S._

Tubo optico 8 pedum.

[Read Nov. 16, 1758.]

h ´ ´´
Initium penumbræ 9 15 18
Initium dubium eclipsis 9 22 24
Certo jam incœperat 9 23 34
Umbra ad mare humorum observata vitro plano cæruleo 9 31 2
Solo tubo optico observata 9 31 29
Vitro flavo observata 9 31 48
Umbra tangit Grimaldum observata vitro plano cæruleo 9 31 20
Solo tubo optico 9 31 50
Vitro plano flavo 9 32 8
Totus Grimaldus tegitur observatus vitro plano cæruleo 9 34 4
Solo tubo optico 9 34 28
Vitro flavo 9 34 47
Umbra ad Tychonem observata vitro plano cæruleo 9 38 25
Solo tubo optico 9 38 42
Vitro flavo 9 38 59
Umbra ad Harpalum vitro cæruleo observata 9 55 6
Solo tubo optico 9 55 35
Umbra ad Fracastorium 9 59 57
Umbra ad Mare Nectaris 10 00 50
Observata vitro flavo 10 1 8
Umbra ad Dionysium 10 5 2
Umbra tangit Mare Tranquillitatis 10 5 50
Umbra ad Mare Serenitatis 10 10 16
Umbra tegit Menelaum observata vitro cæruleo 10 11 4
Solo tubo optico 10 11 29
Vitro flavo 10 11 50
Totum Mare Fœcunditatis tegitur 10 18 39
Umbra tangit Mare Crisium vitro cæruleo observata 10 22 52
Solo tubo optico 10 23 12
Vitro flavo 10 23 29
Umbra ad Proclum 10 23 33
Possidonius totus tegitur 10 23 50
Totum Mare Serenitatis tegitur 10 24 36
Totum Mare Crisium ab umbra tegitur 10 30 27
Plato tegitur vitro cæruleo observatus 10 31 26
Solo tubo optico 10 31 48
Vitro flavo 10 32 4
Obscuratio maxima 10 55 40

EMERSIONES.

h ´ ´´
Plato emergit observatus vitro flavo 11 19 5
Solo tubo optico 11 19 31
Vitro cæruleo 11 19 50
Aristarchus emergit 11 21 3
Gassendus incepit emergere observatus vitro flavo 11 25 36
Observatus solo tubo optico 11 25 52
Observatus vitro cæruleo 11 26 11
Gassendus totus extra umbram 11 28 2
Schicardus incipit emergere 11 45 44
Totus extra umbram 11 47 10
Totum Mare Humorum extra umbram 11 46 50
Menelaus extra umbram 11 55 36
Mare Serenitatis extra umbram 11 59 46
Tycho extra umbra observatus vitro flavo 12 00 33
Solo tubo optico 12 00 52
Vitro cæruleo 12 1 14
Incipit emergere Mare Crisium 12 8 31
Totum Mare Crisium extra umbram 12 16 28
Finis eclipsis 12 28 26

Observatio hæc peracta é cœlo claro; umbra autem terræ ita diluta erat, ut maculæ in ea conditæ satis dignoscerentur.

CV. _Singular Observations upon the_ Manchenille Apple. _By_ John Andrew Peyssonnel, _M. D. F.R.S. Translated from the_ French.

[Read Nov. 16, 1758.]

THe cruel effects of the tree called Manchenille are known to all the world: its milk, which the savages make use of to poison their arrows, makes the wounds inflicted with them mortal. The rain, which washes the leaves and branches, causes blisters to rise like boiling oil; even the shade of the tree makes those who repose under it to swell; and its fruit is esteemed a deadly poison. I was informed, as a very extraordinary thing, that a breeding woman was so mad as to eat three of them, which did her very little harm; and this was looked upon as a miracle, and a proof of the surprising effects of the imagination and longings of women with child.

But here is a fact, which will scarce be credited by many persons, who have frequented these Islands: which I declare to be true.

One Vincent Banchi, of Turin in Piedmont, a strong robust man, and an old soldier, of about forty-five years of age, belonging to the horse, was a slave with the Turks eleven years, having been taken prisoner at the siege of Belgrade. He was overseer of my habitation towards the month of July of the year 1756. He was one day walking upon the sea side, and seeing a great number of apples upon the ground, was charmed with their beautiful colours, and sweet smell, resembling that of the apple called d’apis: he took and eat of them, without knowing what they were; he found they had a subacid taste; and having eaten a couple of dozen of them, he fill’d his pockets, and came home, eating the rest as he came. The Negroes, that saw him eat this cruel fruit, told him it was mortal; upon which he ceased to eat them, and threw away the rest.

About four in the afternoon, _viz._ an hour after this repast, his belly swelled considerably, and he felt as it were a consuming fire in his bowels. He could not keep himself upright; and at night the swelling of his belly increased, with the burning sensation of his bowels. His lips were ulcerated with the milk of the fruit, and he was seized with cold sweats; but my principal Negro made him a decoction of the leaves of a _Ricinus_[157] in water, and made him drink plentifully of it, which brought on a vomiting, followed by a violent purging; both which continued for four hours, during which it was thought he would die. At length these symptoms grew less; and my Negroes made him walk, and stir about by degrees; and soon after they were stopped. Rice-gruel, which they gave him, put an end to all these disorders; and in four-and-twenty hours he had no more ailments nor pain; the swelling of his belly diminished in proportion to his evacuations upwards and downwards, and he has continued his functions without being any more sensible of the poison. We see by this, that the effects of the poison of the Manchenille are different from those of the fish at Guadaloupe, which I mentioned.

Dec. 2. 1756.

CVI. _Abstract of a Letter from Mr._ William Arderon, _F.R.S. to Mr._ Henry Baker, _F.R.S. on the giving Magnetism and Polarity to Brass. Communicated by Mr._ Baker.

[Read Nov. 16, 1758.]

Dear Sir,

FOR some time past I have been making experiments on the magnetism of brass, and amongst many pieces that I have tried, find several that readily attract the needle; but whether they have had this property originally, or have received it by hammering, filing, clipping, or any other such-like cause, I cannot yet determine.

I have a very handsome compass-box made of pure brass, as far as I can judge: the needle being taken out, and placed upon a pin fixed properly in a board, and clear of all other magnetics, the box will attract this needle at half an inch distance; and, if suffered to touch, will draw it full 90 degrees from the north or south points; and I think those parts of the box marked north and south attract the strongest. The cover of the box also attracts the needle nearly as much as the box itself.

As to your supposition, that iron may be mixed with the brass, I do not know; but I have been informed it cannot be, as brass fluxes with a much less degree of heat than iron, and iron naturally swims on fluid brass. Besides, many of the specimens of brass I have tried were new as they came from the mill, where they were wrought into plates, and I presume were not mixed[158]; yet these I have given the magnetic virtue to, when they had it not; and some pieces of brass, which naturally attract the needle, seem to the eye as fine a bright yellow as any other, and are as malleable as any I ever met with.

Pieces of brass without any magnetic power, by properly hammering and giving them the double touch, after Mr. Mitchel’s method, I have made attract and repel the needle, as a magnet does, having two regular poles: and I now send you one such piece of brass, which I have thus made magnetical. You will also receive a couple of needles, which I made myself after the late Zachary Williams’s method, and a little stand whereon to place them, the better to shew how this magnetic bar attracts and repels the needle when properly applied; for it must be noted, that in making these experiments it is necessary to employ a very good needle, about 3-½ inches long, well and tenderly set, and not covered with glass.

You will observe, when you try this bar, that the same poles repel each other, and the contrary poles attract; which proves this piece of brass to be indued with true magnetic virtue and polarity. However it must be noted, that though the same poles repel each other, yet, like natural magnets, in contact, or nearly so, they attract each other; therefore when you would shew the repelling power of this brass bar, you must not bring it nearer the needle than ²⁄₁₀ of an inch.

Magnetic brass does not attract iron, not even the least particle, so far as I can find: whether this is owing to the weakness of magnetism in the brass, or to some other cause, I don’t pretend to know.

I have tried to infuse magnetic virtue into several pieces of copper, lead and pewter; but all my endeavours have not been able to make them attract the needle at all. Indeed, when I have held a piece of pewter, that I have tryed to make magnetical, to the needle, the needle would tremble, but not approach the pewter.

I send you another piece of brass, whose either end attracts either of the poles; this I have infused the magnetic virtue into, and can at any time, so as to attract and repel the needle; but, like steel that is set a low blue, it loseth that polarity in a few hours; which may arise for its being too short for its weight, or from its different temper of hardness or softness.

A third piece I also send you, which with all my endeavours I cannot make attract the needle in the least; and yet I can perceive no difference between the appearance of this piece and that of those which do.

Would some ingenious man pursue these experiments, perhaps we might have needles made of brass to act as strongly as steel ones do, which would have the advantage of being less liable to rust at sea than steel ones are.

But my whole design was to shew, that brass is by no means a proper metal to make compass-boxes of, or to be employed in any instrument where magnetism is concerned. For as it is demonstrable, beyond all contradiction, that some brass is found endued with a power of attracting the magnetic needle; that other pieces are capable of receiving it either by accident or design, (let it be from its being mixed with iron, or any other cause whatever) brass must be a very improper metal for compass-boxes, as it may occasion many sad and fatal accidents.

Norwich, Octob. 20th, 1758.

It is well known, that brass has been sometimes found to affect and disturb the magnetic needle; but, to give magnetism and polarity to brass, has not, that I have yet heard, been before attempted. I therefore have taken the liberty to lay the above account before this Royal Society, and have also brought the pieces of brass mentioned therein, which have been thus made magnetical.

H. Baker.

London, Nov. 15. 1759.

CVII. _An Account of the_ Sea Polypus, _by Mr._ Henry Baker, _F.R.S._

_To the Right Honourable the_ EARL _of_ MACCLESFIELD, President _of the_ Royal Society.

[Read Nov. 23, 1758.]

My Lord,

I now return the marine animal your Lordship did me the honour to recommend to my examination; which I find to be a species of one kind of the Sea Polypi, mentioned by naturalists; but I think not very accurately described.

The kinds of Sea Polypi are understood to be,

_First_, The Polypus, particularly so called, the Octopus, Preke, or Pour-contrel: to which kind our subject belongs.

_Secondly_, The Sepia, or Cuttle-fish.

_Thirdly_, The Loligo, or Calamary. And each of these has its different species and varieties[159]. The ancients add the Nautilus; and some sorts of Star-fish might perhaps be not improperly ranged among them.

All of the first kind have eight arms, placed at equal distances round the head; below the arms are two eyes, and the body is short and thick.

The Cuttle-fish, and the Calamary, have each of them ten arms; of which eight are shorter ones, tapering gradually to a point from the head, where they all rise, to their extremities: the other two (frequently called Tentacula) are three or four times as long, perfectly round, slender, and of an equal thickness for above two thirds of their whole length; then spreading into a form nearly like that of the shorter arms. Great numbers of _acetabula_, or suckers, are placed somewhat irregularly on each of the shorter arms, and on the spreading parts of the Tentacula, where some of the suckers are a great deal larger than the rest.

The body of the Cuttle-fish is broad and flat, having within it a broad friable white bone; that of the Calamary is a sort of cartilaginous case holding the intestines, of a roundish oblong shape, furnished with two fins, and having within it a thin transparent elastic substance like Isinglass.

_G. Edwards delin AD. 1758_ _J. Mynde sc._]

The mouth of the Pour-contrel, Cuttle-fish, and Calamary, is placed in the fore-part of the head, between the arms, having an horny beak, hard and hooked like a parrot’s, which some writers call the teeth. The eyes of them all are nearly in the same position.

As the subject under examination resembles in some particulars all the above kinds of Polypi, this short account of them may, it is hoped, render the following description of it the more intelligible: and with the same view, Mr. George Edwards, Fellow of the Royal Society, has been so obliging as to make drawings of the animal itself, in four different positions, and of the natural size; which drawings are herewith presented to your Lordship.

Our Polypus is of the Pour-contrel kind, and I believe of that species called Bolytæna; which is said to have a musky smell; but if ours had such a smell, the spirits wherein it lies have taken it quite away.

In the drawing [_See_ TAB. XXIX. _Fig._ 1.] is shewn the anterior part of this animal, which has much the appearance of a Star-fish. Here are eight arms about three inches in length, united at their roots, and placed circularly at equal distances in the same plane, which has a considerable sinking towards the center. These arms diminish from their rise to their extremities, and end exceedingly small. Near the head they are quadrilateral, but the under-side contracting gradually to an edge, they become towards the ends trilateral. On the upper side of each arm are two rows of _acetabula_, or suckers, standing in a beautiful order, as close as they can well be placed, and beginning from the center of all the arms. These suckers are perfectly circular, with edges flat on the top, and a round cavity in the middle of each. They are largest in the widest part of the arm, and lessen as the arm diminishes, till they become so small as hardly to be discernable. It is very difficult to tell their number: I counted as far as fifty in a row, but am certain there are many more; and I don’t imagine the eight arms have so few as a thousand on them. They rise some height above the surface of the skin; and wherever they are not, the skin of the arms (unless on the under-side) is granulated like shagreen[160].

As in the other kinds of Polypi the mouth is placed between the arms conspicuously enough, I expected to find it so in this; but the spirits had contracted it so much, that I could discern no opening at all where I thought the mouth must be; and therefore could not say, with assurance, that the mouth was placed there. Under this difficulty I applied to Sir Hans Sloane’s most valuable collection of natural history in the British Musæum, where I found several species of this kind of Polypi, and amongst the rest a small dried specimen of the same species as ours, and a much larger one in spirits, of a species that comes very near it.

This large specimen afforded the information I stood in need of: for though here also the mouth was closed, and the beak drawn down into the center between the arms, so as not to be seen at all; yet, by the help of Dr. Morton and Mr. Empson, I had the satisfaction to see the mouth opened, and the beak in the same situation, and of the same form and substance, as in the other kinds of Polypi. Having gained this knowledge, by applying the point of a bodkin, I easily felt the beak in our Polypus; but in so small a subject it cannot be brought to view without dissection, which is the reason it does not appear in these drawings.

_Fig. 2._ represents the Polypus so placed as to shew the situation of the eyes and the form of its body, and also in what manner the arms are turned back in the specimen before us; but we may suppose them thus disposed merely in the act of dying, and that when alive they are moveable in all directions.

On that side of the body opposite to the eyes, and which therefore may be termed the belly-part, there appears a transverse slit or opening in the skin, not in a strait line, but a little semicircular; from the anterior part whereof a tube or pipe proceeds, about one third of an inch in length, smaller at the extremity, where it opens with a round orifice, than at the base, and reaching to within a small distance of the arms. As both the Cuttle-fish and Calamary have a pipe nearly in the same situation, though somewhat different in figure, through which they occasionally discharge an inky liquor, and some writers say the fæces also, it is probable the pipe in this animal may serve to a like purpose; and as the body of the Calamary is included in a case, the slit across the body of this animal shews its belly part to have also a sort of case, though on its back there is no separation as in the Calamary.

Out of the aforesaid slit or opening a bag issues with a very slender neck, extending towards the tail, and enlarging gradually to its end. This bag is above half the length of the body, and appears like another body appendant thereto. I should be intirely at a loss concerning this bag, did not some passages in Mr. Turberville Needham’s curious observations on the milt vessels of the Calamary enable me to form some conjectures about its use.

Having dissected several Calamaries on the coast of Portugal, without the least indication of milt or roe, and consequently without knowing which were male or female, he was much surprised (about the middle of the month of December) to find a new vessel forming itself in an obvious part, and replete with a milky juice. This was an oval bag, in which the milt vessels formed themselves gradually, the bag unfolding as these framed and disposed themselves in bundles. Before that time he had observed two collateral tubes, which are alike in both sexes; but a regular progress in the expansion of the milt-bag and formation of the milt-vessels had not presented itself before. Those tubes till then appeared open at one extremity, much resembling the female parts of generation in a snail, but did not terminate in a long oval bag extending in a parallel with the stomach more than half the length of the fish, as he found them afterwards when the milt vessels that filled the whole cavity were ripe for ejection. The same ducts without the bag are found in the female also, perhaps for the deposition of the spawn. Vid. _Needham’s Microscopical Discoveries, cap._ v.

It appears from this account that the male Calamary (at a certain time of the year only) has a bag wherein the milt-vessels are contained, and that the female has no such bag. Since therefore the bag of our Polypus is found in the same situation as that of the Calamary, (which is also a kind of Polypus) we may suppose it to be the milt bag, and that our Polypus is a male, taken at a time when the milt was ready for ejection. In the dried specimen at the British Museum, and also in the other specimens, there is the same opening, with the pipe that rises above it towards the arms, but not the least appearance of the bag in question: they are therefore probably females, or if males, were caught before such bag was formed.

_Fig. 3._ presents another view of this Polypus, its arms extended circularly with their under-sides next the eye, and the body so disposed as to shew the transverse opening _a_, the oval bag issuing therefrom _b_, and the pipe rising upwards towards the arms _c_.

_Fig. 4._ shews the Polypus with its transverse opening and the pipe rising therefrom, but without the oval bag; it is figured thus by Rondeletius and Gesner, and the specimen at the British Museum has also this appearance. It is here shewn with the arms extended forwards. K is a magnified figure of one of the _acetabula_, or suckers; of which there are two rows on each arm of this Polypus, as before described.

Mr. Needham, in his description of the suckers of the Calamary, (which he had many opportunities of examining whilst alive, and whose mechanism is probably the same as in those of our Polypus) informs us, “that the action of the suckers depends partly on their shape, which, when they are extended resembles nearly that of an acorn-cup, and partly upon a deep circular cartilaginous ring, armed with small hooks, which is secured in a thin membrane something transparent, by the projection of a ledge investing the whole circumference about the middle of its depth, and not to be extracted without some force. That each sucker is fastened by a tendinous stem to the arm of the animal: which stem, together with part of the membrane that is below the circumference of the cartilaginous ring, rises into and fills the whole cavity when the animal contracts the sucker for action. In this state whatever touches it is first held by the minute hooks, and then drawn up to a closer adhesion by the retraction of the stem and inferior part of the membrane, much in the same manner as a sucker of wet leather sustains the weight of a small stone.” Vid. _Microscopical Discoveries_, p. 22.

M shews one of the cartilaginous rings armed with small hooks, of its real size. The ring this is drawn from was taken out of a large sucker of a larger Polypus, and is presented herewith.

By these suckers the Polypus can fix itself to rocks, and prevent its being tossed about in storms and tempests; but their principal use must undoubtedly be to seize and hold its prey: and to this purpose they are most admirably adapted; for when they are all applied and act together, unless the Polypus pleases to withdraw them, nothing can get from it whose strength is insufficient to tear off its arms. Something like these suckers is found by the microscope in the minute fresh water Polype, whereby it is able to bind down and manage a worm much larger and seemingly stronger than itself[161]. In like manner the _stella arborescens_ (which may also be called a Polypus), though it has not suckers, yet by the hooks along its arms, and the multiplicity of their branchings, which have been counted as far as 80,000, it can, by spreading its arms abroad like a net, so fetter and entangle the prey they inclose when they are drawn together, as to render it incapable of exerting its strength: for however feeble these branches or arms may singly be, their power united becomes surprising. And we are assured nature is so kind to all these animals, that if in their struggles any of their arms are broken off, after some time they will grow again; of which a specimen at the British Museum is an undoubted proof; for a little new arm is there seen sprouting forth in the room of a large one that had been lost.

It is evident from what has been said, that the Sea Polypus must be terrible to the inhabitants of the waters, in proportion to its size (and Pliny mentions one whose arms were thirty feet in length); for the close embraces of its arms and the adhesion of its suckers must render the efforts of its prey ineffectual either for resistance or escape, unless it be endued with an extraordinary degree of strength.

Sea Polypi are frequent in the Mediterranean: but Mr. Haviland of Bath, to whom we are obliged for this, which is of a different species, thinks it came from the West Indies, where it is called a Cat-fish. That like it in the British Museum also came from thence.

As the Polypus I have endeavoured to describe is much contracted by lying long in spirits, and dissection would destroy a specimen well worth preserving, I hope to be excused if this account should be found deficient in several particulars, or chargeable with some mistakes.

Permit me the honour to be,

My LORD,
Your Lordship’s
Most humble and obedient Servant,
H. Baker.

Strand, Nov. 23d, 1758.

CVIII. _A Description of the fossil Skeleton of an Animal found in the Alum Rock near_ Whitby. _By Mr._ Wooller. _Communicated by_ Charles Morton, _M. D. F.R.S._

[Read Nov. 23, 1758.]

IT is in this rock, that the Ammonitæ, or Snake-stones, as they are commonly called, are found, which have undoubtedly been formed in the _exuviæ_ of fishes of that shape; and though none of that species are now to be met with in the seas thereabouts, yet they in many particulars resemble the Nautilus, which is well known. The internal substance of those stones, upon a section thereof, appears to be a stony concretion, or muddy sparr. Stones of the same matter or substance, in the shape of muscles, cockles, &c. of various sizes, are also found therein, and now and then pieces of wood hardened and crusted over with a stony substance are likewise found in it.

_Part of the Fossil Skeleton of an Animal as it appeared on and united to the Allom Rock near_ Whitby, _Jan. 3. 1758_.

a. a. _&c. The Ammonitæ or Snake Stones_.

_J. Mynde sc._]

Many naturalists have already observed, that among the vast variety of extraneous substances found at several depths in the earth, where it is impossible they should have been bred, there are not so many productions of the earth as of the sea; and it appears by the accounts of authors both ancient and modern, that bones, teeth, and sometimes entire skeletons of men and animals, have been dug up or discovered in all ages, and the most remarkable for size commonly the most taken notice of. In the first particular this skeleton will most probably appear to have belonged to an animal of the lizard kind, quadruped and amphibious; and as to its size, much larger than any thing of that kind ever met with or found in this part of the world; though, from the accounts of travellers, something similar is still to be met with in many of the rivers, lakes, &c. of the other three.

When the annexed drawing thereof was taken January 5, 1758. [_See_ TAB. XXX.] there remained no more of the _vertebræ_ than is therein expressed; that is, 10 between D and F, and 12 between G and H: but when it was first discovered, about 10 years ago, they were compleat; and there was besides the appearance of what was then thought to have been fins, near the back part of the head at A, the same as appeared further backward at E, when this design was made. The _vertebræ_, &c. now wanting having been either dug up by curious persons, or washed away by the violence of the waves at high water, and the accidental beating about of stones, sand, &c. during that time; the water covering this skeleton several feet at high water in spring tides; the cavities in the rock still remaining as in the design.

The substance of the bones, with their _periostium_, on the covered or under side, in most parts remains intire, and their native colour in some places in a good measure preserved, and the teeth with their smooth polish plainly to be discovered. Part of the mandible near the extremity was covered with a shelf of the rock about three inches thick; which being cut away and removed, both the mandibles appeared under it compleat, with the teeth of the upper and under one, plainly locking or passing by each other. These appeared to be of the _dentes exerti_ or fang kind, as well as all the others in the narrow part of the mandible, and further backwards they were not observed. From this ledge or shelf the mandible towards B is single, and appears to be the upper one of the living animal; and from the head not being exactly in the line of the body, that part has been inverted, or quite turned over, and the body itself, as appears from the transverse processes of the _vertebræ_, lies on the right side. There appears one row of teeth only on each side of the mandible, and they are about ¾ of an inch asunder.

The mandible B A, the _cranium g h_, and the _vertebræ_ from D to F, were attempted to be taken up whole; but the bones being rendered extremely brittle, and the rock in which they were fixed being a brittle blackish slate, with joints or fissures running in every direction, would not hold together: the whole therefore fell in many pieces, the _vertebræ_ in the joints only, which makes them easy to join together again, and besides shows very plainly the transverse and spinal processes thereof, with the foramen in the latter for the spinal marrow. It was now that a piece of the _os femoris_, about four inches long, shewed itself in the sparry concreted substance at E, together with a piece of the _os innominatum_, to which it had been articulated or joined. This, with what has been before remarked, will sufficiently prove this to have been an animal of the quadruped, and probably, from the shape of the cranium peculiar to fishes, of the amphibious kind. At the same time many pieces of the _costæ_ or ribs, as broke and crushed up against the _vertebræ_, were plainly visible. The cavities of all the bones were filled with a substance, which appeared the same as the rock itself; and the substance on each side the _vertebræ_, as they laid, was a mixture of sparry concreted matter with that of the rock itself, which is a blackish slate. The animal, when living, must have been at least 12 or 14 feet long. And the dimensions of the whole, or particular parts of the skeleton, may be measured from the scale annexed thereto.

This skeleton lay about six yards from the foot of the cliff, which is about sixty yards in perpendicular height, and must have been covered by it probably not much more than a century ago. The cliff there is composed of various _strata_, beginning from the top, of earth, clay, marle, stones both hard and soft, of various thicknesses, and intermixed with each other, till it comes down to the black slate or alum rock, and about 10 or 12 feet deep in this rock, this skeleton laid horizontally, and exactly as designed. The probability, that this cliff has formerly covered this animal, and extended much more into the sea, is not in the least doubted of by those that know it. The various _strata_, of which it is composed, are daily mouldering and falling down; and the bottom, being the slaty alum rock, is also daily beat, washed, and wore away, and the upper parts undermined, whence many thousand tuns often tumble down together. Many antient persons now living, whose testimony can be no way doubted of, remember this very cliff extending in some places twenty yards further out than it does at present. In short there is sufficient evidence, that at the beginning it must have extended near a mile further down to the sea than it does at present; and so much the sea has there gained of the land.

These are the principal facts and circumstances attending the situation and discovery of this skeleton; which from the condition it is in, and from the particular disposition of the _strata_ above the place where it is found, seem clearly to establish the opinion, and almost to a demonstration, that the animal itself must have been antediluvian, and that it could not have been buried or brought there any otherwise than by the force of the waters of the universal deluge. The different _strata_ above this skeleton never could have been broken through at any time, in order to bury it, to so great a depth as upwards of 180 feet; and consequently it must have been lodged there, if not before, at least at the time when those _strata_ were formed, which will not admit of a later date than that above-mentioned.

_P. S._ In the xlixth vol. page 639, of the _Philosophical
Transactions_, an animal is described by Mr. Edwards, which was
brought from the Ganges, and resembles this in every respect. He
calls it _Lacerta (crocodilus) ventre marsupio donato, faucibus
Merganseris rostrum æmulantibus_.

PHŒNICIAN Coins.

_J. Mynde sc._]

CIX. _A Dissertation upon the_ Phœnician _Numeral Characters antiently used at_ Sidon. _In a Letter to the Rev._ Thomas Birch, _D. D. Secret. R. S. from the Rev._ John Swinton, _M. A. of_ Christ-Church, Oxon. _F.R.S._

[Read Dec. 7, 1758.]

Reverend Sir,

HAVING, by the assistance of the Palmyrene numeral characters, lately made a discovery, which may perhaps hereafter be of considerable service to chronology; I could not longer defer, though now deeply engaged in other matters, communicating it to the Royal Society. Nor will the memoir containing this, I flatter myself, be deemed altogether unworthy the attention of that learned and illustrious body. For, unless I am greatly deceived, it will bid fair to ascertain, with a sufficient degree of precision, the Phœnician dates of several antient Sidonian coins, one of which was struck above a century before the birth of CHRIST, hitherto utterly unknown; and evince the notation of the Phœnicians, at least those of Sidon, when they first appeared, to have been extremely similar to, if not nearly the same with, that of the Palmyrenes.

I.

A small brass coin of Sidon[162], now in my possession, exhibits on the reverse three Phœnician letters, that form the word SIDON, over the prow of a ship, the usual symbol of the city wherein it was struck. This coin, which is in good conservation, I formerly[163] published and explained. The characters however in the exergue, which I could then make nothing of, were not with sufficient accuracy described. This has induced me to transmit you another draught of the same medal, wherein proper care has been taken to remedy that defect. The two first of those characters, though somewhat imperfect, appear manifestly enough to be _Schin_ and _Tzade_; as the former occurs on the Palmyrene[164] marbles, and the latter on several very valuable[165] Phœnician coins. The others so nearly resemble the numeral characters of the Palmyrenes, that they may undoubtedly be considered as pointing out to us a date. Which if we admit, the _Schin_ and _Tzade_ will seem to be the initial letters of the words צה שנת, THE YEAR OF SIDON, or IN THE YEAR OF SIDON; as the elements _Pe_ and _Schin_ apparently denote שנת פסח, THE PASCHA OF THE YEAR, or IN THE BEGINNING OF THE YEAR, on the reverse of the famous Samaritan coin of Bologna, published by Sig. Bianconi[166] not many years since. Nor can the phrase, THE YEAR OF SIDON, or IN THE YEAR OF SIDON, intimating the year of the proper æra of that city, be looked upon as repugnant either to the Jewish or Phœnician genius; a similar expression having been used, both in their writings[167] and on their coins[168], about the time that the Phœnician medal before me was struck, by the Jews. That the first of the numeral characters here stands for TWENTY, we may infer from the correspondent one of the Palmyrenes, to the form of which it is by no means unlike. This will likewise be confirmed by the dates preserved on other Phœnician coins, which will be immediately produced. The next, denoting a lesser number, and not representing FIVE, which we find always expressed by minute right lines on the Sidonian medals, must indubitably occupy the place of TEN. The six following strokes, after what has been just observed, will be acknowleged to add SIX to the foregoing numbers; so that the inscription in the exergue will no longer remain a mystery, the whole only importing, IN THE YEAR OF SIDON XXXVI.

II.