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ENOTHERA WHITNEYI, p. 340, is evidently Godetia grandiflora, Lindl. Bot. Reg. 28, t. 61; but it may retain this name in Enothera on account of the old E. grandiflora, Ait.

LEWISIA BRACHYCALYX, Engelm. MSS.: foliis spathulatis vel sublinearibus; scapo haud articulato nudo (ima basi tantum bibracteolato); calyce herbaceo decussatim 4-sepalo (sepalis ovatis) petalis 7 - 9 cuneato-obovatis 2-3-plo brevioribus; staminibus 10-15; stigmatibus 5-7 stylo breviore. W. New Mexico, Dr. Newberry. Fort Whipple, Arizona, Doctors Coues and Palmer; and Utah, Dr. Brewer, in herb. Engelmann. Plant with the habit of Talinum pygmæum and Calandrinia acaulis, but necessarily associated with Lewisia on account of the number of the sepals and the dehiscence of the capsule (circumsissile at base); yet too closely related to Calandrinia. Flowers less than an inch long, "fragrant; petals white with purple veins." Cotyledons incumbent; those of L. rediviva are figured in Bot. Beechey, probably incorrectly, as accumbent. G. Englemann.

LEUCOTHOE DAVISIE, Torr. MSS.: Euleucothoe; foliis ellipticooblongis utrinque obtusis tenuissime serrulatis breviter petiolatis ; racemis terminalibus laxifloris folia longe superantibus; sepalis ovatolanceolatis cum bracteolis bracteisque subscarioso - albidis; antheræ loculis bimucronatis. — Nevada Co., near Eureka, California, Miss N. J. Davis; one out of a fine and beautifully prepared collection of plants recently made by her in that district. In mode of growth, inflorescence, and flowers, this interesting new Western representative of the genus accords with Leucothoe proper (as characterized in Gray's Manual), except that the racemes are terminal and much surpassing the leaves. These are one and a half or two inches long.

PHACELIA (EUTOCA) HYDROPHYLLOIDES, Torr. MSS.: humilis, e rhizomate ramoso repente multicaulis; foliis ovatis subrhombeis oblongisve obtusis inciso-paucidentatis lobatisve nunc lobis infimis fere discretis lyratis utrinque nitenti-sericeis, petiolo laminam æquante caule et inflorescentia hispidulis; cyma terminali parva conferta ; corolla cærulea campanulata calyce paullo longiore, appendicibus latissimis maximis; genitalibus exsertis; placentis 6-8-ovulatis; capsula oblonga acuta calycem adæquante. Ebbett's Pass, and near Lake Tenaya, 8-9,000 feet, Brewer. Open woods along the trail of the Yosemite, from 8,000 down to 5,000 feet, Bolander. Stems 3-6 inches high.

DRAPERIA, Torr. Nov. Gen. Hydrophyllearum.

Calyx e sepalis 5 angustissimis. Corolla tubuloso-infundibuliformis, limbo 5-lobo. Stamina 5, inæquilonga; filamentis inappendiculatis corollæ tubo inæqualiter adnatis. Stylus filiformis, apice bifidus; stigmata capitellata. Ovarium biloculare: ovula in loculis gemina, anatropa, ex apice fere placenta nerviformis pendula. Capsula globosa, subdidyma, membranacea, loculicida, valvis a dissepimento utrinque dispermo solutis. Semina loculos replentia, facie concava medio carinata. Herba multicaulis, humilis, sericco-pilosa; caulibus diffusis basi lignescentibus; foliis oppositis ovatis penniveniis integerrimis longius petiolatis; cymis scorpioideis nudis tenuiter pedunculatis; floribus ebracteatis confertis pallide purpureis. Genus inter Namam et Phaceliam quasi medium insigne.

DRAPERIA SYSTYLA, Torr. Nama systyla, Gray in Proceed. Amer. Acad. 6, p. 37. California, Lobb.; in the Yosemite Valley, Brewer, Torrey, Bolander; Northern California, A. Wood; Nevada Co., Miss N. J. Davis. This genus, particularly interesting as a transition from the Hydrolea to the proper Hydrophyllacea, is dedicated to Professor John W. Draper of New York, author of a Treatise on the Forces which produce the Organization of Plants, and of other distinguished physiological and philosophical works. It was first detected by Mr. Lobb (station not recorded), and briefly characterized by Dr. Gray as a Nama, without examination of the interior of the ovary of the fragment sent to him by Dr. Hooker. With the corolla and nearly the androecium of that genus it combines the gynæcium and seeds of Phacelia proper, except that the ovary and pod are two-celled, and the opposite leaves are peculiar. The embryo has not been found, even in apparently full-grown seeds. J. Torrey.

LEPTURUS BOLANDERI, Thurber, MSS.: pusillus ; culmo basi vaginato superne folio solitario lineari instructo; ligula elongata acuta; vaginis laxis striatis; spica solitaria basi vaginata; spiculis unifloris ; glumis transversis subæqualibus flore vix longioribus; paleis subæqualibus, inferiore basi breviter barbata apice eroso-dentato longius aristata. — Culmi 1-5-pollicares, sæpius ad nodum solitarium geniculati. Folium semipollicare. Palea inferior scabra, costa valida in aristum palea ipsa dimidio breviorem producta. Very distinct from L. paniculatus, which is also found in California. Dry gravelly soil, Russian River Valley, California, Bolander. G. Thurber.

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Five hundred and eighty-fifth Meeting.

September 10, 1867. ADJOURNED STATUTE MEETING.

-

The PRESIDENT in the chair.

The President called the attention of the Academy to the recent decease of Dr. J. Mason Warren and Dr. James Jackson, of the Resident Fellows; of Jeremiah Day, former President of Yale College, of the Associate Fellows; and of Sir William Lawrence, Augustus Boeckh, and Michael Faraday, of the Foreign Honorary Members.

The following paper was presented :

Upon the Logic of Mathematics. By C. S. PEIRCE.

PART I.

THE object of the present paper is to show that there are certain general propositions from which the truths of mathematics follow syllogistically, and that these propositions may be taken as definitions of the objects under the consideration of the mathematician without involving any assumption in reference to experience or intuition. That there actually are such objects in experience or pure intuition is not in itself a part of pure mathematics.

Let us first turn our attention to the logical calculus of Boole. I have shown in a previous communication to the Academy, that this calculus involves eight operations, viz. Logical Addition, Arithmetical Addition, Logical Multiplication, Arithmetical Multiplication, and the processes inverse to these.

Definitions.

1. Identity. ab expresses the two facts that any a is b and any

b is a.

2. Logical Addition. a+b denotes a member of the class which contains under it all the a's and all the b's, and nothing else.

3. Logical Multiplication. a, b denotes only whatever is both a and b. 4. Zero denotes nothing, or the class without extent, by which we mean that if a is any member of any class, a+ 0 is a.

5. Unity, denotes being, or the class without content, by which we mean that, if a is a member of any class, a is a, 1.

6. Arithmetical Addition. a + b, if a, b = 0 is the same as a + b, but, if a and b are classes which have any extent in common, it is not a class.

7. Arithmetical Multiplication. a b represents an event when a and b are events only if these events are independent of each other, in which case ab a, b. By the events being independent is meant that it is possible to take two series of terms, A1, A2, A3, &c., and B2, B2, B3, &c., such that the following conditions will be satisfied. (Here x denotes any individual or class, not nothing; Am, An, Bm, B, any members of the two series of terms, and A, ΣB, (A, B) logical sums of some of the A's, the B's, and the (A, B)'s respectively).

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From these definitions a series of theorems follow syllogistically, the proofs of most of which are omitted on account of their ease and want of interest.

Theorems.

I.

If ab, then b = a.

II.

If ab, and b = c, then a = c.

III.

If abc, then bac.

IV.

If a + b = m and bcn and a + nx, then m+ c = x.

Corollary.

cal addition.

These last two theorems hold good also for arithmeti

V.

If a + b c and a' + bc, then a a', or else there is nothing not b.

This theorem does not hold with logical addition. But from definition 6 it follows that

No a is b (supposing there is any a)

No a' is b (supposing there is any a')

neither of which propositions would be implied in the corresponding formulæ of logical addition. Now from definitions 2 and 6,

Any a is c

.. Any a is c not b

But again from definitions 2 and 6 we have

Any c not b is a' (if there is any not b)

.. Any a is a' (if there is any not b)

And in a similar way it could be shown that any a' is a (under the same supposition). Hence by definition 1,

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aa' if there is anything not b.

Scholium. In arithmetic this proposition is limited by the supposition that b is finite. The supposition here though similar to that is not quite the same.

If a, bc, then b, ac.

VI.

VII.

If a, bm and b, c n and a, n = x, then m, cx.

VIII.

n =

If m, then u, vx.

b and a +mu and anv and a + b = x,

IX.

If m + n = b and a, mu and a, n = and a, b x, then u + v=x.

The proof of this theorem may be given as an example of the proofs of the rest.

It is required then (by definition 3) to prove three propositions, viz.

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