the ultimate ratio which the fum of the given power of the fines bears to a known power of the radius.

Having proceeded fo far as the 6th power the law of continuation became evident; fo that, should any problem in mathematical philofophy require it, we may proceed as far as we please in fumming the powers of the fines. The law is this,

Make a fraction whofe denominator is the index of the given power, and its numerator the fame index, diminished by unity, and multiplied by the fquare of the radius; by this fraction multiply the fum of the next but one lower power, and we have the fum of the given power. Thus ift, the fum of the 1ft power of the fines

is=rr, or the fquare of the radius
2d, fum of the 2d, power or fquares is

=rrx by the arch of 90°.

3d, fum of the 3d, power or cubes is
rr of the 1ft, or=r

4th. fum of 4th powers is rr of the 2d
or=rx by the arch
5th, fum of 5th. powers is

8

or = r2

of 90°.
rr of the 3d,

rr of the 4th

90o.

rr of the 5th,

By Demon ftration.

By Infinite
Series.

or=rx by the arch of 90°.

&c. &c.

Should your leifure permit you to give any attention to
this fubject I shall be glad to fee you furnish a demonftra-
tion for the 3d, or any fubfequent cafe abovementioned.
I am, Sir,

Your moft obedient humble fervant,
DAVID

RITTENHOUSE.

:

Index

Index Florae Lancaftrienfis, auctore HENRICO MUHLEN

BERG, D. D.

Dedicated to the Philofophical Society at Philadelphia, by the author.

GENTLEMEN,

Read Feb.

18th, 1791..

I

Did myself the honour to lay before the Philofophical Society a Specimen Florae Lancaftrienfis, fome time in the year 1786, collected from actual obfervations from 1780.

Since that time, I have had an opportunity of adding fome fupplements, and I now make bold to fend you an index of fuch plants as I could find, after the strictest search, growing either wild or cultivated in, or near, Lancaster. The whole number is very near 1100. All fuch plants as I never found growing wild, but are imported from other countries, or even from other American States, I have marked with a t.

If I found no name in Linne's fyftem, I took it from other works lately printed, or from Doctor Schreber's letters, with whom I have opened a correspondence. He is profeffor Botanices et Hiftoriae naturalis at Erlangen, editor of the Genera Plantarum Linnaei, and, without doubt, one of the first botanifts in Europe. If I could find no name by these means, I was forced to make one myself adding N. S. until better informed by abler botanists. I repeat my former wifh, that fome of my learned countrymen would join in botanical researches, and send in their Floras, for perufal or publication, to your Society, fo that by gathering the Floras of the different States, we may have a general Flora of the United-States, drawn from

good

good and certain obfervations. May I be fo happy as to find your approbation in this fecond fpecimen! I have the honour to remain, with great respect,

Gentlemen,

Your moft Obedient

Humble Servant,

HENRY MUHLENBERG.

Lancaster, Nov. 17th, 1790.

Catalogus librorum quibus ufus eft auctor hujus indicis. 1.Linnaei system, a vegetabilium, curante Murray, 1784. 2. Linnaei fpecies plantarum et genera plantarum, curante Reichard

3. Linnaei genera plantarum, curante Schreber Vol. I.. 1789.

4. Marshalli Arbuftrum. 1785.

5. Walteri Flora Caroliniana. 1787.

6. Aitoni Hortus Kewenfis. 1789.

7. Wangenheim von amarikamipt folzartan. fol. cum figur. 1787.

8. Miller Gardner's dictionary, et cet.

Pleniorem plantarum defcriptionem, cum calendario et ufu medico et oeconomico, brevi tempore v. D. daturus..

Claffis