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 AN ESSAY 
 
 THE MEANS AND IMPORTANCE 
 
 UVTRODIJCIIVG THE IVATURAL SCIEWCES 
 
 THE PAMILTf-LIBRARY, 
 
 DIFFUSING THE ELEMENTS OF GEOMETRY 
 
 THE PLAN OF THE POPULAR EDUCATION: 
 By D. jjMcC URDY, 
 
 LATE A CLERK IN THE U. S. PENSION OFFICE. 
 
 SECOND EDITION, IB(rt>ROVED. 
 
 WASHINGTON: 
 
 BLAIR AND RIVES, PRINTERS. 
 
 1842. 
 
4-J^^- \ 
 
 CAJORI 
 
 Entered, according to act of Congress, in the Clerk's office of the District of 
 Columbia, by Dennis McCurdy. 
 
 Note. — An error in the orthography of this author's name was uninten- 
 tionally admitted into the former edition of this Essay. The remarks made 
 in connection with that error were intended for another occasion. 
 
''' Lhli.^ 
 
 A 
 
 ; o / 
 
 RECOMMENDATIONS. 
 
 The writer of the following discursive remarks served the republic during 
 twenty-two years, die et nocte, as a teacher, in word and sign ; commissioned 
 and caressed by the people long before the strength of the young men had 
 gone forth from the halls of the muses to disseminate literature and science 
 in the land. Then, in 1834, after a suspension of two years or more, 06- 
 •casioned by disease, the effect of his application, he was employed as a clerk 
 in the service of the government ; and, in 1840, notified, with others, that 
 from the 28th of February, 1841, his service should not be required. The 
 last ten years have presented to his view, unsought, shades of human charac- 
 ter deeper than those which he had been accustomed to contemplate ; there- 
 fore, with incredible pleasure, he returns to the pursuits of Science, because 
 he feels within himself the more cordial palpability of the benign radiance 
 beaming from the beauty of her truths. 
 
 Prom many flattering testimonials furnished to him by citizens of this Dis- 
 trict, in orde/ to favor his application for a school in the city of Baltimore, he 
 thinks it proper to select for publication here the letter received from the 
 offic ^ where he had the honor to serve ; for the public service is honorable in 
 ilself; but the manner of obtaining « place may be antithetical, and the fervid 
 ambition, incapacity, &nd catastrophe of Phaeton are recorded for our instruc- 
 tion. 
 
 " WASHINGTON, July 30, 1841. 
 " We hereby certify that Mr. Dennis McCorby has been a clerk in the 
 Pension Office for several years, and is now dropped in consequence of the 
 decrease of the business. During his continuance in office, we have neces- 
 sarily acquired a knowledge of his attainments ; and do most earnestly and 
 unhesitatingly recommend him to the Commissioners of the Public Schools 
 in Baltimore, as possessing high and eminent qualifications as an instructor. 
 
 " GEO. W. CRUMP, Chief Clerk. 
 
 " JOHN D. WILSON, Clerk. 
 
 •« HENRY H. SYLVESTER, Clerk. 
 
 '• WM. S. ALLISON, Clerk. 
 
 ♦• FRENCH S. EVANS, CUrk. 
 
 " DANIEL BROWN, Clerk. 
 
 " JAMES L. EDWARDS, 
 
 *' Commissioner of Pensions." 
 
 The appointment in Baltimore was made in favor of a worthy gentleman, 
 who had the greatest minority of votes at a preceding election for teacher. 
 And, on application to be reinstated as a clerk, the President of the United 
 States made the following endorsement of the papers submitted to him on our 
 behalf; 
 
 „M 
 
6 
 
 dured their scorn, until by pertinacity he drew the current 
 of public opinion into his ship's path over the previously un- 
 explored deep. Upon men without fortune has always de- 
 volved the labor of instructing the world. 
 
 Such is the example which we propose to ourselves in 
 urging the improvement of common school education, by the 
 infusion of the elements of geometry. For this purpose, a 
 plan of a Manual and Chart is proposed : a description of 
 which may be seen in a subsequent part of this Essay. But, 
 in order to place the means of acquiring the knowledge of 
 science in the way of adults of both sexes, we here present a 
 prospectus of a Magazine, which may be imitated by men of 
 science in all practicable places. A periodical publication, 
 commencing with the elements of geometry and arithmetic, 
 and containing a well-arranged and complete course of 
 mathematics, would be read, in many instances, by persons 
 who could never be persuaded to undertake the subject by 
 means of the books used in schools ; especially if the illus- 
 trations should be so simplified as to supply the want of living 
 teachers. Such reading, exclusive of the practical use of the 
 sciences, would contribute to neutralize the deleterious ef- 
 fects of works of fiction, ultra politics, and all that exuber- 
 ance of fanciful matter by which the morals of the age are 
 relaxed, and the line traversed in every direction, which dis- 
 tinguishes right from wrong. The reports of the many 
 violations of law and order would not be read for amusement, 
 which is a certain patronage of the wrong-doer ; the brightest 
 parts would not be employed in depicting the vices and fol- 
 lies of men with a playful coloring of their enormity, from 
 which the fascinated reader glides, among the puppets of the 
 author, into those errors which he finds so elegantly described. 
 The rule of right, upon which the eye of the freeman should 
 ever be fixed, is not laid down in such productions. The 
 false impression is not effaced from the reader's mind, that 
 he is not accountable to God or his country until after he 
 has entered his or her service by special agreement, and 
 signed a pledge, or made profession to that effect. If the 
 author is friendly to religion, he must disguise that. To 
 acknowledge the claim of the Redeemer to the love of our 
 race, is an insuperable barrier to that friendship. His regard 
 for good morals also wants the rule — " To love one's neighbor 
 as himself." Hence such writings do not, in any instance, 
 reprove the reader for faults, nor afford any light for the cor- 
 rection of errors ; which, after all, are mere negative utilities. 
 And thus failing in the first part of the rule of improvement — 
 
<• Cease to do evil," they can have no pretensions whatever to 
 the second part — "Learn to do well." Having, therefore, 
 neither positive nor negative goodness, the ahernative is evil; 
 and, under these views, we cannot be of the number of those 
 who would regret the loss of reputation, as such, to writers of 
 this class, by a voyage from Europe ; because, in our humble 
 opinion, that loss would rather be the recovery of sound dis- 
 cretion to a portion of the people, who had been, to some ex- 
 lent, deluded by the misapplication of great talents — a reduc- 
 tion of over-rated worth to its par value. 
 
 There is also another object to be achieved by the diffusion 
 of the exact sciences, (if our people will be gratified while 
 the Genius of our institutions weeps, and Modesty shrinks 
 from the scene,) namely : to make the amateurs of indecent 
 exhibitions intimate with the properties of the spiral curve ; 
 that they may, at least, be able to criticise, with fastidious ac- 
 curacy, the circumgyrations of all young dancers, patented at 
 the courts of dukes and princes, as one of the means of gov- 
 erning their disfranchised canaille^ whose return to sound 
 discretion these magnates might have some cause to dread ; 
 but who are entirely useless to our people in the administra- 
 tion of their affairs, either private or public. Useless, indeed, 
 and inefficient for any purpose of good, but greatly prohfic 
 as the seed of evil ; because the arfist, corrupt at heart, and 
 ready to sell his country for sordid gain, exhibits for sale 
 multiplied copies of such indelicate exposures ; and, in the 
 interior of his impure shop, finishes the work of shame un- 
 disguised, to which he will invite the ripening debauchee, 
 disgracing humanity, and breaking down the last impediment 
 which obstructs the decline of virtue. Where, do you in- 
 quire, are the seats of such abominations? Tamper with 
 the crime in your theatres, and you will find the copies every 
 where in the streets of your cities. Their favorite [jaunts 
 will be near the legislative halls, dishonoring the precincts of 
 the sacred fanes of human liberty, without the remedy of 
 law, or a vindicator of the public sanctity. For how will 
 your courts be able to convict or punish in the copy that 
 which you have sanctioned and applauded in the original? 
 Citizens ! will you make the fine arts the medium of cor- 
 rupting your morals ? Will you abdicate your rational forms 
 of government, and be led by low sensations unbecoming the 
 supreme dignity of the inheritance of your philosophic and 
 Christian fathers? Will you borrow your taste, as a people, 
 from the kennels of paganism, perpetuated as it is in the 
 unamended condition of a large majority of the populace of 
 the old world ? 
 
s 
 
 THE WASHINGTON MAGAZINE, 
 
 AND 
 
 JUNIOR citizens' GUIDE TO SCIENCE, 
 
 BY D. MgCURDY & CO., 
 
 Will be issued monthly, at $b per annum, payable in advance, making 
 an annual volume of nearly 600 octavo pages, with a plate of the appropriate 
 diagrams to accompany each number. 
 
 It will contain a regular course of Mathematics, commencing with the 
 Elements of Geometry and Arithmetic, in separate departments, and followed 
 by Plane Trigonometry and Algebra, the art of constructing Logarithms, 
 Surveying and practical Navigation, Geometry of Solids and Mensuration, 
 Spherical Trigonometry, Conic Sections, and Astronomv, Carpentry, the 
 Mechanics' Price Book, and Plans of Modern Buildings, Architecture, 
 Civil Engineering, Mechanical Philosophy, &c., &c. It may be remarked 
 that several of these and other branches differ only in name and applica- 
 tion; that the general principles are common to many of them, and that 
 the labor of acquiring the knowledge of the course is less than may be 
 generally supposed : the principal hinderance consists in the impatient haste 
 to pass over the first principles with the absurd question, "What use is 
 there in that r' 
 
 The most approved authorities will be consulted, and the named Editor's 
 long experience in teaching will be put in requisition for the benefit of those 
 citizens who have to contend with the crude materials of Nature in organiz- 
 ing the habitable world. It is not intended to pass with a light saltation from 
 one branch of the tree of natural knowledge to another, but to investigate the 
 permeant elements in their course through the trunk, branches, flowers, and 
 fruit ; and to exhibit in the foreground whatever tends more immediately to the 
 increase of human skill and power. Each particular branch will be based 
 upon a sufiicient number of theorems duly demonstrated, and illustrated by 
 the necessary problems to meet the several cases which those theorems may- 
 embrace ; but where different branches rest upon the same theorems, these 
 latter will not be repeated. 
 
 In order to consociate amusement with instruction, the leading principles 
 and rules of grammar, rhetoric, and logic will be inserted ; and a chapter 
 on the art and forms of keeping accounts; also a succinct historical account 
 of the progressive improvements in the sciences, and a brief biograp^hy of 
 eminent mathematicians, will be introduced at proper intervals in the work ; 
 and, that all classes of readers may be gratified, popular descriptions will be 
 given of the natural and other curiosities contained in the National Gallery ; 
 also a regular account of the most important transactions of the National 
 Institution, recently established in the city of Washington. 
 
 The Washington Magazine will be a family instructor of the cheapest 
 kind ; and a useful companion for the retirement of young men preparing 
 themselves for high distinction in the national mart. Therefore, to extend 
 the advantages of science, the remittance of S20 from any college, academy, 
 village, military or fire company, community or association, in the same let- 
 ter, will be received in full for live copies of the Magazine; and in the same 
 ratio for a greater number. The postmaster certifying the enclosure will 
 make us responsible. The postage must be paid, or the letter franked. 
 
 Remittances may be made, in good money, to D. Clagett, Esq., merchant, 
 Washington, D. C. ; Benjamin Hailowell, Esq., Alexandria, D. C; or Alex- 
 ander Yearley, Esq., Baltimore, Maryland. 
 
AN ESSAY. 
 
 The desire to diffuse, or rather to accumulate, knowledge, 
 has recently received a simultaneous impulse from many 
 points ; and the best talents in the world are concentrating 
 their thoughts on the metropolis of this Union, to render it 
 the Emporium of Science. Here the lines converge ; the 
 march is onward, and no power at this time can retard its 
 speed. The wisdom of men will consist in regulating and 
 directing the momentum to the safest and happiest results, 
 in promoting the contentment and industry of the people, 
 and in preserving the due equilibrium of power by equal 
 education, so as to ensure the faithful execution of the laws. 
 We shall just notice a few of the most prominent indications 
 of the desire referred to above, in order to show, in the out- 
 set, the increasing importance of diffusing the elements of 
 science more equally among the people. 
 
 Indications of the desire to diffuse knowledge : 
 
 1. In the action of Congress with regard to the bequest of 
 Mr. Srnithson. 
 
 2. In the establishment of the National Institution at 
 Washington. 
 
 3. In the action of the friends of popular education in 
 some of the States, resulting from, the disclosures made in 
 the last ceyisus. 
 
 1. The Congress of the United States have assumed a 
 trust, in the case of the Smithsonian bequest, under which 
 they are pledged to establish, in the city of Washington, an 
 institution for the diffusion of knowledge among men. To 
 this undertaking, divested of any evil tendency, we wish 
 
10 
 
 ail possible celerity and success. It may be rendered use- 
 ful; but, like the Grecian gift, it may introduce treason 
 into the citadel of our liberties, by teaching us to love gifts, 
 to depend upon contingencies, and be the less blessed of 
 the two cases — namely, that of the receiver and that of 
 the giver. Our own resources properly employed are suf- 
 ficient for our wants. Giving to this testator's views the 
 best construction, the bequest is a high compliment to 
 our institutions, and to us a memento to persevere in that 
 moderation which is the true characteristic of a people who 
 are willing to do right — who require only the light^ not the 
 compulsive power of the law. In this view of the case, the 
 trust, it must be admitted, is worthy of a free people, and 
 their representatives are in every respect adequate to its ex- 
 ecution. From the assumption of this trust, however, as 
 well as from the annual appropriations for the support of 
 their military academy, and the many excellent documents 
 which are sent out from the halls of legislation at public 
 expense, it is plain that Congress do not mean to admit any 
 constitutional or other restriction of their power to diffuse 
 knowledge. This is a gratifying disposition in the Legis- 
 lature, to which we may hereafter advert in the course of 
 these remarks. 
 
 But with respect to the species of knowledge to be dif- 
 fused under the trust, there are various opinions. It is con- 
 tended, in this case, that no kind of knowledge is proper 
 which our citizens owe to themselves or their children ; for 
 this we should furnish at our own expense. This view, it 
 will be perceived, excludes every branch of useful know- 
 ledge ; because all such we owe to ourselves and our chil- 
 dren, and more as a reigning people than we should as a 
 subject people. Besides, we owe to the world as much as 
 any other member of the family of nations, under the great 
 Father of the universe. Is not our portion of the common 
 inheritance equal to that of any other, nay, better than others? 
 Has he not assigned to us from his table a Benjamin's mess? 
 
11 
 
 Shall we then insult the Fountain of Light by admitting only 
 a few rays, jnst sufficient to cause obliquity of vision, and 
 qualify us to take delight in short-lived and local scenes? 
 The people owe it to themselves to repel all suggestions to 
 restrict or circumscribe their pursuit of knowledge ; and 
 every man should extend his sphere as far as may comport 
 with the means and time he can honestly devote to its ac- 
 quisition. Knowledge is the true lever of equality which 
 can elevate the ponderous mass, and as such we are anx- 
 ious to recommend it to all. We also believe that Congress 
 may constitutionally erect an observatory at the nation's 
 expense, and found a national institution embracing the 
 ample range of the sciences and arts ; that they ought to do 
 so forthwith ; and that the honor and dignity of the republic 
 are compromised in the delay. 
 
 2. Another indication of the desire to diffuse knowledge 
 is clearly developed in, the establishment of an association 
 at the city of Washington, in May, 1840, under the denom- 
 ination of the National Institution. The object of this in- 
 stitution is to promote science and the useful arts. The 
 number of its members already exceeds six hundred, and 
 the increase is constant. The initiation fee is five dollars, 
 and the annual contribution is the same amount. Its mem- 
 bers are resident, corresponding, and a few honorary. The 
 President of the United States is called the patron of the 
 institution ; it has a president, the Hon. Joel R. Poinsett ; 
 vice president, Peter Force, Esq. ; twelve directors, of which 
 six are elected by the institution, and the other six are the 
 members of the cabinet at Washington ; namely, the four 
 Secretaries, the Postmaster General, and the Attorney General. 
 The directors on the part of the institution, at present, are 
 the Hon. Levi Woodbury and Hon. William 0. Preston, 
 Senators; Col. J. J. Abert, chief of the topographical en- 
 gineer bureau ; Col. Joseph Totten, chief of the engineer 
 bureau ; A. O. Dayton, Fourth Auditor of the Treasury ; 
 and Com. L. Warrington, one of the Commissioners of the 
 
12 
 
 Navy Board. Francis Markoe, jr., Esq., is the correspond- 
 ing secretary; G. R. Barry, Esq., recording secretary; and 
 Dr. H. King is the curator of the institution. The subjects 
 of research are divided into eight classes, namely, astron- 
 omy, geography, and natural philosophy ; natural history ; 
 geology and mineralogy ; chemistry ; the application of sci- 
 ence to the useful arts ; agriculture ; American history and 
 antiquities ; literature and the fine arts. Besides the nume- 
 rous specimens of geology, mineralogy, and other objects 
 of natural history collected by the members, the association 
 has in charge the immense variety of curiosities sent home 
 by the exploring expedition. The correspondence of this 
 institution already extends to all parts of the world, and its 
 present condition is highly pleasing and prosperous. Its * 
 depository is the new national gallery — one of the rarest 
 specimens of architecture, and said to be the most spacious 
 room in the United States. We propose to give, in the 
 fixture numbers of our contemplated magazine, a fuller ac- 
 count of the transactions of this institution, and a classified 
 description of the astonishing collection of the productions 
 of nature and art, civilized and savage, placed in charge of 
 the curator of the institution. Dr. King. 
 
 3. The last indication of the growing desire to diffuse 
 knowledge which we shall here notice, is the action of the 
 friends of science in the States, impelled by the disclosure 
 resulting from the statistics collected in the taking of the 
 last census. It appears that many of the people are desti- 
 tute of all literary knowledge, and large masses of the com- 
 munity receive but a very limited education. This action 
 will be the more effectual, because it carries the remedy to 
 the seat of the disease. When the people undertake to do 
 their own work, it will be done. The lark, in the fable, 
 thought it unnecessary to remove her young from the wheat - 
 field while the farmer depended on neighbors or friends for 
 his reaping ; but when he concluded to begin by himself 
 on the morrow, she exhorted her little brood to provide for 
 
13 
 
 their future safety. There will be no more delay in this dif- 
 fusion of knowledge, since the proper agents are to be em- 
 ployed. We are, however, constrained to express the hope, 
 that the basis of whatever system they may adopt will em- 
 brace the" elements of geometry, as being necessary for the 
 support of any superstructure which can avail in erecting 
 the minds of the people. 
 
 Astronomy^ which leads the whole train of the sciences^ is 
 dependent on the elements of geometry. — The National In- 
 stitution, as appears from the preceding aiTangement, has 
 very properly placed astronomy at the head of its list of the 
 natural sciences. This is the study which first inspired 
 man with the spirit of inquiry, and for the sake of which 
 the subordinate sciences have been intensely prosecuted. 
 Dedit homini sid)lime os. In its modern improvements, 
 the application of its principles descends even to regu- 
 late the weights and measures in the hands of the mer- 
 chant. It sweeps the cerulean expanse with the telescope, 
 watches the motions of the heavenly bodies, and collects 
 its data from a thousand stations at a thousand dates; 
 adding to the number of celestial discoveries such revolu- 
 tions and phases as the planets and satellites exhibit from 
 any peculiarity of the observer's position with regard to the 
 ecliptic, or highway of the distant worlds. " And never," 
 says the Hon. Joel R. Poinsett, late iSecretary of War, and 
 senior director of the National Institution, in his excellent 
 discourse delivered at its first anniversary, " never has been 
 commenced a monument to the glory of science and humsui 
 intellect more sublime than that of which astronomy is now 
 laying the foTindation." "Shall we not add," continues 
 Mr. P., " one stone to the structure? Will we expose our- 
 selves to be denied our just title of a moral, intelligent, and 
 enlightened people, by refusing to inscribe the United States 
 of America among the names of the civilized nations of the 
 earth, which will be found engraved upon the columns of 
 this magnificent temple? Are we not a navigating and 
 
14 
 
 commercial people ? Does not our flag float on every sea, 
 and visit every accessible region of the world ? And shall 
 we not have our national observatory, our astronomical 
 archives, and our celestial ephemeris [day-book]? Shall we 
 any longer leave our navigators exposed to the disgrace of 
 acknowledging that, without the astronomical ephemerides 
 published in Europe, they could not with safety navigate 
 distant seas? I hope not." And so, to echo the pious as- 
 piration, we all hope. Nevertheless, whatever praise may 
 be due to the noble science of astronomy, no practical 
 utility can accrue to the community from it, without a pre- 
 vious knowledge of geometry, upon the principles of which 
 astronomy must be superinduced. It is, therefore, our in- 
 tention, in the humble office of pioneer, to co-operate with 
 the magnanimous designs of this excellent institution, by 
 disseminating the elements of science more generally among 
 the people; without which they must come unprepared, 
 though invited, to the intellectual feast. 
 
 The same elegant writer and true patriot, in speaking of 
 the science of technology, or the union of scientific knowl- 
 edge with mechanical skill, instances the case of James 
 Watt, LL.D., who invented the steam-engine, the most per- 
 fect of human devices, " a present," so called, " from science 
 to the arts." And the case of Peter Dolland, who accom- 
 plished through this Union, what even Newton had given 
 up as unattainable. Mr. P. here alludes to Dolland's equa- 
 torial instrument for correcting the errors of refraction in alti- 
 tude, of which the inventor gave to the Royal Society some 
 account in 1779. The discourse here alluded to furnishes 
 an outline of what a National Institution ought to be : it has 
 been distributed gratuitously, and may be had, we believe, 
 at the Patent Office. No intelligent citizen should omit the 
 perusal of this valuable pamphlet, or the subscription of his 
 name to the National Institution therein defined and recom- 
 mended. 
 
 A Magazine of the natural sciences^ for the family library , 
 
15 
 
 should co7nmence loith the elements of arithmetic and geom- 
 etry. — We have thus briefly adverted, ad res gestas, to the 
 things done, or contemplated to be done, at Washington, 
 and in some of the States, for the diffusion of knowledge 
 among men. We desire to point the telescope of consider- 
 ation to the hill of science in the far, elevated, serene 
 distance; that the junior multitude, whom we would rouse 
 to application, may have some idea at least of the direc- 
 tion in which we propose to lead them, by means of the 
 magazine which, with the aid of the friends of science, 
 we propose to publish. But as in the outset of every jour- 
 ney some preparation is demanded, so likewise shall we be 
 required to make provision for the way. It is, however, 
 peculiar to this line of march, that, in collecting the materi- 
 als, our progress will be facilitated ; and the elements which 
 we collect will be no impediment to the celerity of our ad- 
 vancement: on the contrary, the more we collect, the more 
 rapidly shall we approach the object at which we aim. Let 
 these remarks be now applied to the acquisition of the ele- 
 ments of geometry and arithmetic, without the aid of which 
 no progress can be made in mathematical knowledge ; and 
 let it be understood that under these two heads are all the 
 elements of the exact sciences contained. 
 
 Geometry and arithmetic should liave priority of alge- 
 bra. — Mathematical reasoning is conducted according to two 
 methods, — one is called the method of analysis, or resolution; 
 the other, the method of synthesis, or composition. Algebra 
 adopts the former of these ; separating the known from the 
 unknown parts of a general proposition; representing num- 
 ber and magnitude by symbols, and descending by a suc- 
 cession of equivalent propositions from the most complex to 
 the most simple form. Geometry adopts the synthetic me- 
 thod, which begins at the simplest elements, and proceeds 
 to the most complex combinations : moreover it seizes upon 
 the matter and forms of extension in their natural state, or 
 represents them exactly in miniature, making the relations 
 
16 
 
 subsisting between lines and angles, superficies and solids, 
 familiar to the senses, which the algebraic symbols cannot 
 do, from their position and other obvious properties and ac- 
 cidents. In like manner is algebra, with all its facilities of 
 substitution, transposition, and elimination, dependent in 
 its application on a previous knowledge of arithmetic : for 
 it must employ the algorithm of numbers ; and since every 
 number is a term of some series, the relations of the given 
 numbers must be understood from the law of the series to 
 which they belong, and from their position in that series, 
 before any advantage can be gained, either in the demon- 
 stration of a theorem, or the solution of a problem, by means 
 of symbolical representations. 
 
 For these reasons, we shall set forth, in the leading num- 
 bers of our magazine, the elements of geometry and arith- 
 metic ; and shall not attempt to generalize or concentrate 
 propositions too hastily, lest we might defeat our most cher- 
 ished hope, viz : that of enlisting in the cause of science the 
 greatest possible number. Our object is to diffuse this 
 knowledge where it is most wanted ; and there is no greater 
 obstacle to this diffusion than the formulae of algebra ap- 
 plied to principles not clearly understood. This doctrine 
 is urged by Sir Isaac Newton, and his great preceptor, the 
 learned Doctor Barrow, who deprecated in this pursuit noth- 
 ing else so much as the too early application of the analytic 
 method, which is chiefly introduced with the use of alge- 
 bra. There is in algebra no elementary principle : why, 
 then, should it precede those studies upon the principles of 
 which it seeks to be engrafted? Can the proportions of 
 numbers or the relations of magnitudes be at all understood, 
 when expressed only by the letters of the alphabet, which 
 have not now, as anciently, a definite numerical value assign- 
 ed to them? The supposition is preposterous. In short, a con- 
 tracted and obscure view of the uses of mathematical know- 
 ledge has been superinduced by the early application of 
 literal arithmetic and the neglect of geometry : hence this 
 
17 
 
 latter subject is scarcely at all estimated in the plan of the 
 popular education ; the seeds of knowledge are withheld 
 from the youthful and vigorous soil, by the malign influ- 
 ence of inveterate habit, which clings to restrictive and dis- 
 criminating rules in the diffusion of knowledge, even in 
 cases where the means are freely and amply provided. 
 
 It is not our intention, however, to decry algebra, or to 
 dispense with the facilities which it affords as the great 
 auxiliary of science. On the contrary, we shall, in the prose- 
 cution of this work, pursuing our research into the latitudes 
 and longitudes, the depths and sublimities of Nature's laws, 
 and their best imitation in the works of art, avail ourselves 
 of all the aids derivable from method^ whether of composi- 
 tion or resolution ; we shall not omit nor overlook the inter- 
 ests of the practical mechanic, or the man of science; but 
 shall introduce, successively, with proper regard to the nat- 
 ural order and mutual dependence of the sciences, those 
 branches or parts of the mathematics which are most pro- 
 motive of public utility. 
 
 The Magazine should explain the terms which involve 
 principles in arithmetic as icell as in geometry. — In the 
 arithmetical department, it will be deemed expedient to 
 define certain words, for the sake of the principles with 
 which they are connected, and to familiarize our junior 
 readers with the particular views under which these ele- 
 ments are to be presented. A variety of facts relative to the 
 doctrines of numbers will be collected under this head, 
 which may occasion some appearance of delay ; but, as in 
 the clearing of a forest for cultivation, many difliculties will 
 be thus removed, and time and labor saved in the general 
 result. In the list of definitions the words will be placed 
 in alphabetical order, for the convenience of reference ; and 
 it may be necessary to refer to several words, in order to un- 
 derstand the principle to be explained, as the words give a 
 mutual explication of each other, and one illustration sheds 
 its hght upon another. 
 
18 
 
 An explanation of the true frinciples of induction^ as appU- 
 cable to arithmetic. — The organization of units into series^ 
 in the formation of numbers, as the fundamental law, has 
 not been regarded by writers on arithmetic in the full ex- 
 tent of its importance ; and the effects of secondary causes 
 have been substituted in its stead. Hence the light has 
 fallen obliquely, and by reflection, upon many useful prin- 
 ciples, and errors have been interwoven with the truths of 
 science in the protracted shades. 
 
 Arithmetic is the science of the numerical series. This 
 is its true definition ; and it will be found useful to con- 
 sider the series of cardinal, ordinal, multiplicative, and 
 distributive numbers, as in grammar. We shall here 
 notice, however, only the series of cardinal numbers, or 
 the numerators of things. In whatever way a relation may 
 occur between required and given numbers, that relation 
 must be investigated along the line of some series. And it 
 may be remarked that all series consist of consecutive sums, 
 differences, products, or quotients ; since these include all the 
 operations of numbers. The series of numbers may all be 
 ranged into two principal divisions : first, the regular, which 
 have their terms increased or decreased by a constant equal 
 quantity ; and, second, the irregular, whose terms are in like 
 manner affected by a variable quantity. Each of these 
 divisions may also be subdivided into classes : and of the 
 regular division, the class of series which have their terms 
 equi-different are called arithmetical; and the class whose 
 terms are continued proportionals, ere called geometrical. 
 Again : of the irregular division are all additions and sub- 
 tractions of simple numbers, where the result is pursued, 
 as it were, by placing the given terms consecutively along 
 the line of the natural series ; also the series of triangular 
 numbers, which, for several years, under the name of lot- 
 teries, has been the medium of the most frightful system of 
 deception and furtive traffic, with the sanction of ignorant 
 or iaterested rulers, and supported by the cloying cozenage 
 
19 
 
 K)Ta venal press. To digress a little into the history of this 
 case, it may be remarked, as a very probable circumstance, 
 that the much boasted plethora of a certain State treasury is 
 occasioned by the revenue derived from the neighboring 
 •cities by the sale of lottery tickets ; from which delusive 
 and criminal traffic the unhappy adventurers can realize 
 no other issue than a succession of disappointments. 
 
 But to return to our subject: To the irregular division 
 belong all the series of squared, cubic, and pyramidal num- 
 bers ; and those series whose terms consist of two or three 
 unequal dimensions, as in the piling of balls, <fcc. In this 
 division are also the series of powers and roots ; but the log- 
 arithmic series of indices or exponents is arithmetical, and 
 belongs to the regular division. Of each division and class 
 there is also an infinite variety of series, and the natural 
 series contains the terms of every other. The decimsd 
 terms 1, 10, 100, 1000, &c.,or their multiples, as high as 
 the ninth inclusive, in reversed order, constitute the organic 
 law of the notation of numbers; and, lastly, every series 
 has its own measuring unit, to which every unit of that 
 series must be equal in dimensions and value. 
 
 In view of all which, it is obvious that the compilers of 
 systems of arithmetic have not assumed the true principle 
 of the science, either in their definitions, rules, or exempli- 
 fications ; that they have marked out no certain path to its 
 perfection ; and, consequently, that the progress of their fol- 
 lowers is caliginous and erratic. In this consequence, the 
 editors who have professed to favor the inductive system are 
 also involved, because they have given no evidence what- 
 ever that they have made the discovery of the general 
 law into ichich leads induction. 
 
 Ttie leading numbers of the Magazine should not contain 
 abstruse questions, — This magazine will contain a complete 
 course of mathematics, and be rendered subservient to the 
 purposes of education in families ; aiding adults in revis- 
 ing their former studies, which may have been passed over 
 
20 
 
 too superficially ; and juniors, by a collocation of questions, 
 under appropriate heads, in mercantile and mechanical arith- 
 metic, adapted to the most ordinary capacities, and refer- 
 red to the proper series in every instance ; so that, among 
 the earliest lessons, the laws of the several classes of series 
 which form the basis of all the modern improvements in 
 the mathematics will be made familiar to the reader. The 
 object of this publication will preclude, at the first, those 
 abstruse investigations in which the sons of science delight 
 to exhibit their skill ; and no question can be admitted into 
 its pages in anticipation of the principles upon which such 
 question is founded. We intend to instruct, not to puzzle 
 our young readers. Our first numbers will be devoted to 
 the elements of science, for the use of that large majority of 
 'ci,tizens who want " the appliances and means'' of improve- 
 ment. It required strength to bend the bow of Ulysses, and 
 detailed practice to perform the problem of the arrow gind 
 ten rings; so skill in, and power over, the elements of 
 science accumulate by continued acquisitions. 
 
 Geometry is at war with the lawless condition of mind and 
 Tnatter. — The elements of the exact sciences cannot be treat- 
 ed in a discursive or amusing manner; on the contrary, the 
 random of fancy is to be arrested, sophistical arguments are 
 to be avoided, and the mental faculties required to submit to 
 the impression of truth unmixed with error. Geornetry deals 
 in realities, rests on admitted principles ; and its process of 
 investigation is the test of philosophical truth. But when 
 these principles are duly cultivated, they afford the most 
 ample means of rational amusement ; proving that reality is 
 more astonishing than fiction, because the diversified opera- 
 tions of Nature are more in number, beauty, and splendor, 
 than the wild imagery of the most fruitful fancy, and the 
 discoveries of science excel all the painted scenery of poetry 
 and romance. Therefore the votary of science is amply re- 
 warded for the time and pains expended in acquiring the 
 radiments of this knowledge, which has to contend for its 
 
^1 
 
 v>ery existence against the gayer allurements of the spume, 
 and the unprincipled drowsiness oi X\iQ sediment ihdX mingle 
 too freely with the better elements in the composition of 
 the world. 
 
 The investigations of the sciences dependent on geometry 
 are limited within the knoion parts of the creation. The 
 line and plummet must never be out of use. — The sublimest 
 effort of these studies is to trace upon the tablet of the 
 human understanding a copy of the plan of the creation, 
 the temple of the Deity ; and to demonstrate the action 
 of its laws from the partial view of them attainable in 
 this remote position in the solar system. They do not 
 carry us out of the limits of God's works beyond the region 
 of light — extra fiammantia mcenia mundi ; nor place us an- 
 terior to the existence of time and the motion of bodies, 
 when space alone existed : because, in the contemplation 
 oi empty space, there is no point with which to compare, no 
 principle of action or motion, and consequently no notion 
 of life and its concomitant enjoyments; therefore, to date the 
 principles of this knowledge anterior to the motion of bodies, 
 except as existing in the wisdom of the Deity, is an absurd 
 abstraction, an ti- geometrical, a mere speculation of infidelity, 
 which never stops at the proper point, and, from its morbid 
 disaffection to the Supreme Ruler and his laws, runs into 
 all those absurd extremes to which geometry accords the 
 most consummate reprobation. 
 
 From the motions of the upper worlds the sciences de- 
 scend to examine the contrivances of art, and to teach man- 
 kind, from the combination of a few simple principles dis- 
 covered in matter, to exert incredible force and ingenuity 
 in the production of the most useful ends. These subjects 
 will claim our future attention ; and having access to the 
 Library of Congress, the Patent Office, and other sources of 
 information, it will be our aim to select such materials as 
 shall be deemed most profitable to our readers, in accordance 
 with our intended plan of conjoining theory with applica- 
 
22 
 
 tion, and of ascending the hill of science, as the guide of a' 
 multitude of our junior co temporaries, by easy, that is, hy 
 regularly graduated steps. 
 
 The popular education is defective with regard to means. 
 — Means are wanted to supply the defect, as far as possible,, 
 in the popular education. There are, it is ascertained, about 
 one hundred colleges in the United States ; and in these a 
 number of students somewhat short of ten thousand are re- 
 ceiving instruction. It may also be inferred, from the late 
 census, that near three millions of persons are of the proper 
 age for instruction in schools. Deducting the ten thousand 
 students from three millions, there will remain two millioa 
 nine hundred and ninety thousand. This immense majority 
 derive no direct advantage from colleges, and very little in- 
 directly; because graduates of this order generally have 
 higher pretensions than that of the humble office of com- 
 mon-school teacher. 
 
 Let us inquire, next, whether the principal academies will 
 make amends for the want of college assistance : of these 
 there may be six hundred in all the States, giving instruc- 
 tion to sixty thousand pupils ; which, deducted from the 
 last remainder, will still leave two million nine hundred and 
 thirty thousand, who either receive no instruction whatever, 
 or so limited a share of it as to leave them unqualified for 
 the duties of statesmen, legislators, judges, magistrates, cler- 
 gymen, physicians, lawyers, military or naval officers, navi- 
 gators, or even masters in any of the subordinate trades and 
 professions. Yet, from this overwhelming majority of un- 
 qualified citizens, the several offices above mentioned must 
 be supplied y or, at least, they are to be the judges who will 
 determine as to the qualifications of those who are to fill* 
 them. 
 
 An uneducated population are the materials of a despotism. 
 — Now, one of two destinies awaits our form of government: 
 either it shall become a despotism, or one of progressive so- 
 cial improvement; dependent, in the latter case^ on the gea- 
 
23 
 
 «ral and more equal diffusion of knowledge. The materials 
 of a despotic government are the millions whose education 
 is so far neglected as to eradicate their sense of equality, 
 who, losing sight of the first general maxim in the Declara- 
 tion of Independence, are ready to admit that they are a sub- 
 ordinate class, and who look to no alternative but a change 
 of masters. The population here described, whose individ- 
 ual ambition can avail them nothing, havCv nevertheless 
 a powerful collective ambition; though they are undis- 
 ciplined in the laws, regard their operation as partial, and 
 abhor their perplexity, yet they have patriotism, swell at 
 the recital of their country's glory, and never forget the 
 name of a successful warrior. They undervalue civil ac- 
 quirements, the operations of which have no sensible effect; 
 but they are all eye, all ear, all heart, to the tumult and os- 
 tentation of war. The consequences of their united action 
 may be these, — that the chieftain who shall lead them out to 
 repel their country's invaders may probably lead them in to 
 remodel the government more agreeably to his will. The 
 machine will be found too complex for the general and his 
 followers, and they will reduce it to the simple form of a 
 general order. The eye of such a populace rests upon the 
 executive, not upon the laws. 
 
 Domestic education is the most available. — Such are the 
 results which must necessarily arise from the neglect of 
 the popular education; and these results, which every 
 good citizen must earnestly deprecate, are not so remote 
 in the long vista of ages as we may supinely think ; the 
 means of prevention cannot therefore be safely delayed. 
 The means of discipline in useful knowledge, to meet 
 the exigence of the case, must be sent home to the resi- 
 dence of every citizen, to the farm and shop, where labor 
 and study may relieve each other, and the common mind 
 be enlightened with science at the least expense either of 
 the endearments which sweeten the springs of domestic fe- 
 licity, or of the necessary labor of the people and their pecu- 
 
24 
 
 niary means. They have neither means nor opportunities 
 to acquire the proper culture at remote and higher institu- 
 tions. 
 
 Equal means of education are necessary to preserve a free 
 government^ and avarice is its antagonist. — There are, as 
 we have remarked, three millions of persons in the United 
 States who should be receiving instruction sufficient to 
 qualify them for any office in the people's gift. Seventy 
 thousand of these derive the form of a liberal education 
 from colleges and higher schools, and two million nine 
 hundred and thirty thousand are comparatively neglect- 
 ed. But who will undertake to affirm that these, the lat 
 ter class, are less endowed with natural intellect than the 
 former? — or that, in their business through life, they less 
 require the aids of literature and science ? The defect of 
 equal and general means for the education of d^ free people, 
 is thus obvious to the least penetrating observer. And this 
 defect tends to create different orders in the community ; 
 to be maintained for a time by art, but finally by an armed 
 aristocracy, as in Mexico, against the common rights of the 
 people, and the provisions of our excellent constitution. It 
 is, however, contended that education makes men more 
 vicious. That must be a bad education, for which we have 
 no desire to provide ; and we also admit that the disparity 
 of a good education leaves a portion of the citizens unpro- 
 tected. But the argument against education, whatever it is 
 worth, is drawn from an equality of ignorance ; and we aver 
 that the equality of intelligence will ever have the same 
 claim to innocence. Besides, intelligence is the characteris- 
 tic of freedom, of which the United States furnishes con- 
 clusive proof. Education should be of that kind that resists 
 the commonest and most pernicious vices. Now, avarice 
 is the first vice of this and every other commercial people. 
 The present generation are suffering the inevitable con- 
 sequences of this passion, which is antagonist to a com- 
 monwealth, and which was unknown to the fathers of the 
 
25 
 
 republic. The accumulation of great wealth is a very 
 ambiguous test of virtue, and one with which the people are 
 never so well satisfied as the security of the possessor 
 might require. Hence, the throes of the country : hence 
 the strong effort of our institutions to defecate the impure 
 principles which have aimed at their subversion. Let go 
 your ill-acquired gains, increase the amount of productive 
 labor, and educate men to moderation, and all will be well. 
 It is commonly charged upon the profligate and the idle, 
 that though property were for once equally divided, another 
 division, and anotlier, would be necessary to preserve the 
 equality. But this consequence is produced by the cove- 
 tous man rather than the spendthrift. The latter relaxes his 
 grasp upon his property, and the former snatches it out of 
 his hand. One is generous and unsuspecting, the other is 
 subtle and rapacious. One would permit a portion of the 
 public property to be in common, the other appropriates to 
 his own use every thing which he is able to seize. If all 
 the wealth which can be acquired from the imbecility of 
 others, or through the error of legislation, be justly yours, 
 why not at once return to the old doctrine, that might is 
 right? 
 
 Legislation requires intellectual discipline and experience 
 of the relations of society. — We would, However, insist on 
 the necessary qualification in our rulers : we say, they ought 
 to be educated men. It is our intention to excite emulation, 
 not envy. Let us examine societ^n its formation : it is the 
 natural condition of man. The writers on international 
 law, to form some basis for an argument, have supposed 
 cases of solitude which have never existed. To talk of 
 rights in the case of one who has abandoned all his rights, 
 and can suffer no wrongs, unless those which he may inflict 
 on himself in his gloomy forest cave, is one of the ex- 
 travagancies of infidelity for which some of these popular 
 writers are remarkable. Man came into society when he 
 came into existence: all men are born equally helpless; 
 
26 
 
 they have a right to life, for the same reason that society- 
 has a right to exist; and, for any thing that appears in 
 evidence in the matter, one is as well entitled to liberty 
 and the pursuit of happiness as another — id est, liberty to 
 seek or pursue his happiness ; — which happiness consists 
 in the enjoyment of that temperature of mind that is ef- 
 fected by the unresisted action of the law of his being. 
 The Creator has willed man's happiness, but the means 
 are not acquiesced in: hence society requires to be con- 
 nected and secured by sanitary regulations. The mem- 
 bers of society are dependent on each other; the whole 
 body is dependent on the Great Original Existence. The 
 machine is a complicated one, and its laws must be in 
 some measure understood by the man whom we could con- 
 scientiously recommend to one of its responsible offices. 
 Therefore, we aver that legislators, judges, and executive 
 officers should understand the structure of the body politic, 
 and also the rights and moral obligations of the whole, as 
 well as the several parts. In short, every human enactment 
 ought to assume to be nothing more than a simple declara- 
 tion of some inference deducible from that general rule, by 
 the observance of which the happiness of the whole and of 
 the parts is promoted. 
 
 For such reasons as these we would rather see men hold- 
 ing responsible offices who have undergone a course of 
 mental discipline ; but we wish all, or a sufficient majority, 
 as many as possible, to He eligible ; therefore we desire to 
 let out the streams of science upon this national husbandry 
 in every possible direction. The available media for the 
 diffusion of knowledge, now in existence, are the colleges, 
 academies, and common schools ; of which the number of 
 the latter bears to that of both the former about the same 
 ratio as three hundred to one. 
 
 Concentration and diffusion of knowledge contrasted, — 
 The concentration and the diffusion of knowledge may 
 be contrasted. High and expensive institutions tend to the 
 
27 
 
 foraier rather than the latter ; the seeds of knowledge, like 
 those of plants, must be disseminated upon the broad basis, 
 not grudgingly, but with a generous hand ; we may add, 
 after the manner of Christianity. The same may be said of 
 eminent works of science. To the Principia of Newton, per- 
 haps, no well-adapted additions could be made; but the 
 general diffusion of this colossal knowledge requires means 
 different from any that have been hitherto employed ; and 
 so likewise of geometry and other natural sciences. Col- 
 leges and academies are too expensive to be placed at all 
 points ; it follows, therefore, that the common schools of the 
 country are the proper media for the diffusion of knowledge. 
 But it may be seiid that these are already effecting the ob- 
 ject contemplated in their establishment. This may indeed 
 be true in numerous cases ; but the object so contemplated 
 may not be commensurate with the increasing necessity of 
 knowledge. We have known certain schools which have 
 been supported liberally, at public expense, for a quarter of 
 a century, in which the progress of science has been stopped 
 at the point which they technically call the rule of three. 
 This may have been the contemplated limit of the object 
 for which such schools were established ; and the practice 
 has corresponded to the design. But such restriction is evi- 
 dently incompatible with the march of mind, as well as 
 with the natural fecundity of the mental soil. Was this the 
 proper method to neutralize the superior advantages of col- 
 legiate instruction, and equalize the acquirements of the 
 citizens in knowledge and power? Certainly not. The 
 case alluded to may, perhaps, be of rare occurrence, repre- 
 senting common schools in their least efficient state. 
 
 The metaphorical tree of knowledge. — The tree of knowl- 
 edge is a familiar figure of speech ; and the departments 
 of science are so commonly denominated branches^ that 
 this term is scarcely regarded as a metaphor, but rather 
 as a literal expression. It would, however, be no disad- 
 vantage to the cause if the figure were carried out more 
 
28 
 
 extensively, so as to embrace the trunk, juices, and fruit; 
 also the condition of the tree, whether as a stunted shrub 
 or a stately fruit-bearer. This would afford a basis for a 
 more vivid illustration of the minute origin, connexion, 
 dependence, and progress of the elements ; showing, from 
 the nature of the fruit, branches, &c., that the seed is of two 
 kinds — number and extension ; that with number alone, only 
 a stunted shrub is produced ; that although the larger and 
 smaller branches of the stately tree partake of the properties 
 of numbers, yet those of extension are so interwoven in the 
 entire texture as to preclude access to the fruit by the skill 
 and power of numbers alone ; that the season of planting 
 and culture ought not to be neglected, and that the soil 
 ought to be duly selected and prepared. 
 
 The elements of science are not acquired at the proper 
 season, and therefore rarely ever. Their light is hid as under 
 a bushel ijBom the mass of the people ; or, to use a continen- 
 tal simile, the channel of their mighty river is closed by the 
 sunken rafts of time-honored erudition, and the deep current 
 passes off on the sinister hand, for the exclusive benefit of 
 a very limited number. Even in the best institutions the 
 quantity of this knowledge is abated, from the want of appro- 
 priate means of acquiring it in the preparatory schools. 
 
 School commissioners and trustees are bound to lay a pro- 
 per foundation for p^iblic instruction. — To incorporate the 
 elements of geometry with the popular education is an 
 indispensable duty, binding upon those who are intrusted, 
 by commission or otherwise, with the direction of this 
 important branch of public affairs. Let your pupils read 
 and become familiar with Euclid's propositions and dia- 
 grams at the proper age, before the opportunity is snatched 
 away from them by procrastination. Consider the results 
 in the cases of Newton and Bowditch, and a thousand 
 others of merely accidental opportunities of reading works 
 of science. The former of these great men was not in- 
 tended for a liberal scholar : he was employed as a helper 
 
29 
 
 on his mother's farm, when Euclid's Elements fell into 
 his hands : he understood the propositions of that author 
 by simply reading them ; they presented to his capacious 
 mind no difficulty ; and from such powerful manifestation 
 of intellect was drawn the motive for sending him to the 
 university. He lived to express his regret that he had occu- 
 pied so much time upon the theories of Des Cartes, and that 
 he did not limit his investigations within the range of the 
 principles of Euclid's Elements, as being the most direct 
 way to his own discoveries. Doctor Nathaniel Bowditch, 
 it is well known, collected the seeds of science, by which 
 he has done honor to his country, and established for him- 
 self an imperishable name, from some volumes which he 
 found in the ship-chandler's office where he was employ- 
 ed. Give to the rising generation the means of perusing 
 the elements of geometry, — this is the defective point for 
 which there is no equivalent study, — and the next age will 
 be illustrious for its Newtons and Bowditches. Then will 
 arise many ornaments of society in this department of knowl- 
 edge. After the manner of Sir Charles Scarborough, they 
 will be able to repeat, in order, all the propositions of 
 Euclid, Archimedes, and other ancient authors ; and, like 
 Sir Isaac, comprehend them without the labor of much 
 study. The eminent mathematician above mentioned was 
 physician to Charles the Second, and his two successors, as- 
 sistant and successor to Doctor Harvey, as lecturer on 
 subjects of anatomy and surgery, and the first who judi- 
 ciously applied mathematics to medical subjects. 
 
 A Chart and Manual of Geometry for the use of schools 
 recommended, — In addition to the magazine herein proposed 
 to be published, as one of the incipient means of diffusing the 
 knowledge of the exact sciences more equally and generally 
 among the people, we have devised a series of charts of the 
 diagrams of geometry, and a manual of the definitions, pos- 
 tulates, axioms, and propositions of Euclid to accompany 
 the charts. This, indeed, is but the seed of a glorious har- 
 
30 
 
 vest of knowledge — worthy of being so esteemed, because of 
 its general diffusion ; and general, because of its congeniality 
 to the operations of every well-regulated human mind. It is 
 our earnest wish to make the impression of the forms of 
 extension precede the work of analysis, and to furnish the 
 corresponding propositions according to the ancient and ap- 
 proved order of the Greek geometer. Any deviation from 
 this arrangement would only open the way for an endless 
 train of pretended improvements, which would no more 
 subserve the purpose of general diffusion than the present 
 refinements of analysis are effecting. 
 
 A description of the Chart and Manual of Geometry <j ivith 
 the title of each^ and the supposed terms of their publica- 
 tion. — The chart No. 1, of the series here alluded to, will 
 contain the diagrams of the six books of Euclid ', exhibit- 
 ing nearly three hundred strong lined figures, on an area of 
 twenty square feet of plate paper laid on cloth, with mould- 
 ing and roller, bound and varnished in the manner of a map ; 
 or, as it will consist of four equal plates, if required, it 
 may be so mounted as to fold in quadruple, without the 
 wood, for the purpose of transportation. The manual 
 No. 1 will contain the definitions, axioms, postulates, and 
 propositions of the same books, wanting the demonstra- 
 tions. The rules for constructing the figures will, however, 
 be given; also the references to the elements preceding 
 each, usually quoted as proofs. 
 
 1. The Chart of Geometry, No. 1, exhibiting the dia- 
 grams of the first six books of Euclid : to accompany the 
 Manual of Geometry, No. 1, containing the propositions and 
 other elements of the same books. Price, $5 per copy. 
 
 2. The Manual of Geometry, No. 1, containing the defi- 
 nitions, postulates, axioms, and propositions of the first six 
 books of Euclid : to accompany the Chart of Geometry, 
 No. 1, exhibiting the diagrams of the same books. Price, 
 not exceeding $3 per dozen. But each copy of the chart 
 will be entitled to one copy of the Manual without extra 
 charge. 
 
31 
 
 It will be recollected that, wherever geometry is taught, 
 the figures are commonly drawn on a blackboard, and ef- 
 feced with the occzision ; losing thus the advantage deri- 
 vable from subsequent reference. If this practice should 
 still be continued in connexion with the chart, the latter 
 will serve as a record of such transactions, and be a use- 
 ful auxiliary in the college hall, as well as a pioneer in the 
 preparatory school. 
 
 Geometry and arithmetic should be taught at (he same 
 time. — It is susceptible of the most satisfactory demonstra- 
 tion that arithmetic and geometry should be simultaneous 
 studies ; it is also well known that the want of a practicable 
 plan, alone, has in many instances prevented this union. 
 It is proper, however, to remark, that the former requires 
 the aid of the latter, rather than the contrary. Geometry 
 depends for the proof of its propositions upon its own ele- 
 ments; whereas arithmetic requires the extended dimen- 
 sions of geometry to illustrate the principles of its opera- 
 tions : this will apply, however, to the case of rectilineal 
 figures. In the theories respecting curves, the excellence 
 of Newton's demonstrations consists in his superlucid ap- 
 plication of the doctrine of infinite series, in which geom- 
 etry requires the aid of numbers ; in the mixed mathe- 
 matics, they are professedly subservient to each other. 
 
 The entire labor of demonstration should not be at once 
 imposed on the learner. — It is reduced to a maxim, that the 
 classification of labor facilitates its performance. The plan 
 here proposed is to omit the demonstrations, but not any of 
 the elementary principles, in the commencement of this 
 study. The first care of the learner should be to know each 
 proposition in order ; its data, or given parts ; its quaesita, or 
 requirements ; to construct the corresponding diagram, or at 
 least to trace it out with a pencil, or otherwise, by its points, 
 lines, and angles ; so that it may be clearly identified \vith 
 the proposition, and the enunciation of the proposition 
 readily given from an inspection of the diagram. These 
 
32 
 
 preliminary exercises, which do not fall far short of the full 
 amount of labor required in this study, may be performed in 
 the family circle, in schools of any grade, by any person 
 who can read, as well as by a learned geometer. Let no 
 time be lost in committing to memory the definitions, 
 axioms, and postulates, since the constant reference to them, 
 at each proposition, will soon render them familiar; read 
 them, and pass on to collect other facts. 
 
 Youth is the proper time for acquiring science, and form- 
 ing an intelligent citize?i. — In order to form the young 
 mind to place within it the organizing principles of geome- 
 try, the burden of this study must be assumed in youth, — 
 that season of life in which the memory is susceptible of 
 defined impressions, and most willingly submits to the im- 
 position of a task ; and these elements must be thus ac- 
 quired, or the demonstrations cannot be efiected ; he shall 
 only make the discovery of his own impotency (for want 
 of knowledge is want of power) who will undertake the 
 latter without the necessary preparation. And unless a man 
 possess the intuitive perception of the truths of science, or 
 receive it by a course of instruction, a morbid ignorance and 
 hesitating uncertainty will embarrass all his thoughts, or a 
 precipitate, turgid, unwarranted assurance will mark his 
 course through life ; one unbalanced passion will control 
 him ; he cannot be an enlightened citizen of this republic ; 
 he will behold the improvements of art with a vacant stare, 
 and be a helpless loiterer in the march of mind. The moral 
 influence of the sciences is not duly appreciated in the af- 
 fairs of this Government, dependent as it is on the popular 
 will. 
 
 Geometry taken in the Euclidian order is not abstruse, 
 and m.ay be learned by any good reader. — If it should be 
 objected that the study of geometry requires maturity of 
 judgment and riper years, it may be answered that the 
 mind, in this case, i^ required to create nothing ; the state- 
 ments and proofs are all furnished ; and the principles, as 
 
33 
 
 well as their application, are as simple and obvious as those 
 of grammar or geography. This latter branch is. indeed, 
 a dependent on geometry, deriving its forms and propor- 
 tions from it, having been successfully and commonly 
 taught in schools only a few years, — that is, since the atlas 
 accompanied the book of descriptions. And as to gram- 
 mar, it is comparatively abstruse and metaphysical ; not a 
 subject of form or extension, and therefore evading the 
 action of the senses ; yet it has been reduced to the ca- 
 pacity of juvenile learners, and made a common study 
 in all the schools. Geometry, taken in the Euclidian order, 
 is easy and natural; an isolated problem, depending on 
 principles wholly unknown to the performer, (and this 
 is what we propose to provide against,) presents an insur- 
 mountable barrier to the tallest intellect ; like the elevating 
 of a block of marble upon a lofty column, which cannot be 
 effected without the intervention of the mechanic powers ; 
 so the difficulties of all such problems are removed by a 
 previous acquaintance with the Elements of Euclid. With 
 a due regard to the progressive abilities of the learner, it 
 may be said that he is prepared to enter upon this study so 
 soon as he can read sufficiently well to exercise his memory 
 on the subject of his lection. 
 
 Teachers may acquire this knowledge from the recitations 
 of their pupils, tcithout any previous study. — To teachers 
 unacquainted with geometry, but who desire to extend 
 their sphere of usefulness, the plan here proposed presents 
 the most favorable opportunity of learning from the recita- 
 tions of their pupils. The frequent rehearsal of the same 
 task through successive classes, with the continual ref- 
 erence to the diagrams on the chart, and to the preceding 
 proofs at each proposition in the Manual, cannot fail to give 
 the impression of this knowledge in the shortest possible 
 time. Many zealous and qualified laborers would thus enter 
 the fields of science ; the laws of Nature would be more fully 
 explored, and made the subjects of general reflection and 
 3 
 
34 
 
 models of action among men. Hence the most beneficial 
 results would accrue to the community from the regulating 
 influence of those perfect examples upon the ardent passions 
 of the young : the moral as well as the intellectual condi- 
 tion of society would be improved, by diffusing the knowl- 
 edge of Nature's laws, of whose operation geometry is not 
 a system of symbolical representations, but rather a min- 
 iature likeness. 
 
 Colleges have a pecuniary interest in the promotion of this 
 preparatory study. — Professors and teachers who have the 
 charge of instructing young men in the mathematical de- 
 partment in colleges and academies are expected to sanc- 
 tion this plan, until a better can be provided, of inculcating 
 an early acquaintance with geometry. Here the materials 
 will be prepared to some distance in advance of the quali- 
 fied demonstrator, who will doubtless find the business of 
 wrangling not only facilitated, but greatly multiplied on his 
 hands, by the immission of eager applicants from the lower 
 schools, where the use of the manual and chart may be 
 adopted. Men are not apt to care for that of which they 
 know nothing, however good it may be in itself; but a par- 
 tial insight into this knowledge will quicken a desire to 
 know more : and there can arise no rational fear that, in the 
 case of the most assiduous student, during the years usually 
 passed at college, the subject of the mathematical sciences 
 will be exhausted. 
 
 Geometry is the law of Nature, as it regards action or mo- 
 tion. — And what is geometry ? let it be asked ; or the study 
 which we are endeavoring to advocate ? To premise, in 
 answer, we remark, that the works of art are imitations, not 
 creations. It may be truly asserted that in mechanics no 
 usefiil discovery has ever been made, even inadvertently, 
 unless where the mind had been previously drawn out, with 
 or without design, into that natural order of thought, which 
 the association of contiguous objects and a succession of 
 generated motions had suggested to it; and that every 
 
35 
 
 ^iseful invention is nothing more than the development of 
 «ome natural law, although the inventor himself may not 
 ^e able to recollect the train of reflections which conducted 
 him to his conclusions. This may serve to show the wis- 
 dom of investigating the causes and effects of things in the 
 way which Nature's laws prescribe, in order that advantage 
 may be taken of those irreversible rules of action, and a 
 helpful co-operation instituted between nature and art. In 
 this consists the truth of the maxim that " knowledge is 
 power." The folly of opposing those rules, or dispensing 
 with them, is set forth in the proverbs of every language of 
 which we have any account ; and every year, in the con- 
 duct of some one or other who spends his time and money 
 in the fruitless search of a motive power inherent in the yet 
 unknown combinations of matter. The laws of motion and 
 forces are here particularly alluded to ; and these pervade 
 the universe. Every part of the creation is in motion ; air, 
 earth, water, fire, light, heat, sound, time, the planets, comets ", 
 and it may be (for the standard of comparison is too great 
 for our comprehension) the whole solar system, all — all are 
 in motion, unvarying, incessant. To these are analogous? 
 the mechanical and chemical actions of the works of art ; 
 and all are subject to the rules of geometry. The study of 
 geometry is therefore an inquiry into Nature's laws. 
 
 A few of the many uses of geometry to a free people enu- 
 merated. — Self-government is the great business of the peo- 
 ple who would be free : this consists in a voluntary conform- 
 ity to th€ best natural and social regulations ; which implies 
 a knowledge in the people of their rights and their duties, 
 their wants and their means of providing for them. To a 
 self-governing people every kind of useful knowledge is in- 
 dispensable, and there is none more prolific of utility than 
 geometry. We have already alluded to the maxim that 
 "knowledge is power;" and besides this, we have it upon 
 the authority of holy writ, that " the soul to be without 
 knowledge is not good." If, therefore , the people of the 
 
United States will preserve their two-fold character of rulew 
 and subjects, and execute the double function in the best 
 manner, the range of their popular education must^ be ex- 
 tended, and the pabulum mentis^ the intellectual nutriment, 
 supplied in proportion to the growing wants of their in- 
 creasing population. To this end every facility should be 
 given to the diffusion of the useful arts, very many of which 
 depend for their origin, progress, and perfection, upon the 
 knowledge of geometry. To enumerate all the uses of this 
 science, would be to evoke the practical operations of the 
 better half of the busy world. Let it suffice to say that it is 
 eminently the science of national defence by land and sea -, 
 of strategy or the marshalling of troops ; of the motion of 
 projectiles and of gunnery; of the building of ships and 
 fortifications ; of navigating the trackless ocean, from the 
 comparison of times and distances, marking the revolutions 
 of the stars and planets. Geometry is the science which 
 teaches the erection and adornment of edifices ; the construe - 
 tion of bridges, mills, machines, and engines ; the distribu- 
 tion and subduing of the glebe ; the graduating and con- 
 ducting of watercourses and railways; the surveying of 
 coasts, shiproads, and rivers, and marking the relative posi- 
 tion of headlands and other stationary objects for the gui- 
 dance of the mariner in view of land. Geometry furnishes 
 all the principles for the fabrication of musical, optical, and 
 nautical instruments, and the means generally of exerting 
 human power successfully against the resistance of matter 
 in all its forms, so as to accomplish with extreme precision 
 the minutest as well as the most stupendous works. 
 
 The study of geometry produces internal order, regulates 
 the morals, and elevates the mind to contemplate the sacred 
 order of the Creator^s works. — Geometry is a moral science, 
 or in closer alliance with the science of morals and religion 
 than any other of the physical sciences. The study of it 
 places within the soul, as it were, a framework of truth, on 
 which to dispose in due order the faculties, affections and 
 
37 
 
 passions, under the control of reason; a trellis for the sup- 
 port and display of the fruits of virtuous principles, hon- 
 orable to the student, and salutary to those who are favored 
 with his worthy example. The diseases incident to a col- 
 lapsed condition of the moral powers find here a remedy ; 
 which powers, so arranged, instead of exhausting their force 
 in mutual collision, severally act their appropriate functions 
 within their destined spheres. The mind is thus led, from 
 its own internal order, to contemplate the sacred order of 
 the Creator's works ; to admire the wisdom of their arrange- 
 ment and the harmony of their several parts ; to exult in 
 deriving its own existence immediately from the Almighty 
 Architect himself. 
 
 Good government is a Divine unity ; and a freeman must 
 either exercise it upon himself or fall into the hands of another 
 master. — It is not, however, pretended, nor can it be be- 
 lieved, that good morals or good government can exist in 
 the absence of religion. Grood government is an indivisible 
 unit, descending from the Blest Supreme, and apportioned to 
 his creatures as they can bear it. It pleases the people of 
 the United States to hold the reins for themselves, by God's 
 grace, not feignedly as usurpers, but as of duty under the 
 most sacred obligations, taking the ultimate responsibility, 
 or to depute the task for a time to chosen men, whose duty 
 it is made to direct the power of the whole equally for the 
 good of each individual : but where self-government is 
 vigorous, external government is the less wanted, and des- 
 potism fails of its object through the want of materials upon 
 which to act. But that man is not free who has lost the 
 government of himself — who has broken loose from his 
 own restraint. The apostolic gradation is the theory of 
 perfect liberty, — faith, virtue, knowledge, temperance, pa- 
 tience, godliness, brotherly kindness, charity : these libera- 
 ting operations are effected by Divine assistance, and not 
 otherwise ; for we cannot know even natural things in their 
 true relations, unaided by the internal illustrations of tlie 
 
wisdom which is from above. Nevertheless, the study oi 
 the natural sciences is here recommended in connexion with 
 that of redemption ; but all history attests that the study of 
 the mathematics preponderates vastly in favor of true and 
 exalted piety. 
 
 The diagrams and propositions of geometry should be 
 every where exhibited, — If, therefore, these elements of the 
 natural sciences be of so very great importance to the 
 people of the United States, why should any rational 
 means of rendering them effective be neglected in the 
 popular education? The public and private service of the 
 country calls for a more general diffusion of this know- 
 ledge. Either, then, let some better means be devised, or 
 let this chart of the diagrams be spread out, and this man- 
 ual of the propositions of Euclid be suspended to it, as the 
 farmer suspends his almanac for the service of the year, 
 in every district school ; in every mechanic's shop, every 
 counting-room, every public, society, or firemen's hall, 
 where young men resort; in the domicil, among the rising 
 youth ; at every military post and naval station ; in the 
 quarters of midshipmen, and petty officers and seamen, when 
 at sea ; and, where it might not be an inappropriate ap- 
 pendage, in the libraries of young lawyers, with the Com- 
 mentaries of Blackstone and Kent. Then will abler ad- 
 vocates enter the forum, furnished with Eu<ilid's invincible 
 style of argument and the strong fulcrum of truth, on 
 which to rest a plea ; the wet seaboy would devote his 
 watch below to the improvement of his nautical skill;. 
 old soldiers, among whom are not often wanting those who 
 would recognise in Euclid a former acquaintance, would 
 take new delight in explaining to their comrades the na- 
 ture and application of his diagrams, reminding them of 
 the long defence of Syracuse, by the science of Archimedes, 
 against Marcellus, the sword of Rome ; the apprentice 
 would tax the leisure of his servitude with the acquisition 
 of this knowledge, which would qualify him to test the 
 
strength, under certain combinations, and overcome the 
 resistance or resist the pressure of the massive materials 
 used in the works of art; and the domicil and school 
 would send forth to the halls of legislation orators more 
 succinct and cogent, statesmen better informed respect- 
 ing the nature of productive labor, the cost of public works, 
 and the economical expenditure of the revenues of the 
 country. 
 
 Let the temperance associations, especially, recommend it 
 to the young, as a substitute for the coarse revelry with 
 which they aflfect to be happy. While reformed men, and 
 those who have always shunned the cup, are uniting their 
 efforts in rooting up the vice of inebriety with all its con- 
 comitants, the soil which they clear will not lie fallow ; the 
 seeds of many evils exist in the human constitution, ready 
 to spring forth in native luxuriance, in the absence of restrain- 
 ing goodness. The active mind of man seeks employ- 
 ment ; and inexperience unfits it for judging of that which is 
 most profitable. If the elements of geometry be dissemi- 
 nated and receive proper culture, the fruit will be perfec- 
 tion in many of the useful arts, and purity and power of 
 self-protection in the details of our institutions. This 
 would produce a salutary action in the body politic, and 
 prevent many of the disorders which disturb the peace of 
 the community. 
 
 Geometry is necessary to complete the education of an 
 American citizen. — It was anciently said that there is no 
 royal road to geometry ; or, in other words, no way to under- 
 stand the laws of Nature without the labor of study : this is 
 now a truism, as much as it was in the days of Ptolemy. Here, 
 however, there is no royalty to desire exemption from labor 3 
 and freemen labor for themselves ; also, in the pursuit of 
 knowledge they seek a treasure beyond the reach of fraud. 
 But, as freemen labor for themselves, so they govern them- 
 selves ; they therefore require the more skill and knowledge, 
 and those in the greatest variety, from skill in the lowest man- 
 
40 
 
 ual operation to knowledge of the highest exercise of the 
 moral functions of man, that they may perform with decency 
 the subordinate duties of life, and be qualified to govern 
 the state in the true spirit of its constitution and laws. 
 The necessity of greater knowledge is an honorable dis- 
 tinction, but the acquisition of it is more honorable ; and 
 the adage Ne sutor ultra crepidam is wholly inapplicable 
 to an American citizen. Let this study, therefore, be in- 
 corporated with the other useful branches in the general 
 plan of the popular education, in order to complete the ba- 
 sis of a system commensurate with the dignity of this 
 form of government, based as it is upon the inalienable 
 rights of man. 
 
 The General and State Governments should patronize the 
 general diffusion of geometry as a military science. — The 
 legislative bodies of the several States, as well as the 
 General Government, should take this subject into their 
 consideration. In the cases of the former, there are, it is 
 believed, no constitutional restrictions ; we are not aware 
 that their authority has ever been questioned as regards pro- 
 viding by law for the education of their citizens, regulating 
 the matter and manner of that education, and dispersing the 
 seeds of knowledge freely among the people. 
 
 If, however, the jealousy of the States should deny to 
 Congress such a construction of the constitution as would 
 empower that body to legislate on this subject, then we 
 would inquire upon what principle they act when they re- 
 serve so many sections of the public lands for the purposes 
 of education ? wherefore do they educate one youth from 
 each congressional district at a military school? and by 
 what authority do they devote their time, as a legislative 
 body, in the trusteeship of the Smithsonian bequest? If 
 Congress may appropriate lands, money, and time, for the 
 diffusion of knowledge, who will deny them the power of 
 direction as to the characterof that knowledge? Geometry 
 is the basis of the knowledge of the most feublime sciences 
 
41 
 
 known to the human mind, and the best adapted to the 
 condition of a free people ; it is therefore indispensable in 
 the contemplation of any institution which such a people 
 might be expected to establish for the diffusion of general 
 knowledge. Moreover, it is that science which is most con- 
 genial with a condition of want and necessity, and most ready- 
 to enter as one of the principal means into any plan for the im- 
 provement of humanity in its rudest state. But it is also a 
 military science, connected with the defence of the country : 
 if it be expedient, therefore, to diffuse the knowledge of 
 field tactics, it is expedient, a fortiori, to diffuse that of 
 geometry, which includes the principles generally that lie 
 at the basis of the art of war. This diffusion, however, can- 
 not be effected, in either part, through the single officer who 
 graduates at West Point : therefore the militia, to whom the 
 country belongs, and who are therefore the proper defenders 
 of it, must remain destitute of this knowledge so necessary 
 to concert of action in the day of trial and peril, or be depend- 
 ent upon the mercenary skill of others for the defence of their 
 property and lives. Again : if it be the province of Congress 
 to provide by law for organizing companies, battalions, and 
 regiments, and designating their commanders ; also for fur- 
 nishing arms, equipments, and rations, — is it not likewise the 
 province of that honorable body to provide for the diffusion 
 of such knowledge among them as will give their united 
 action the desired effect ? 
 
 To the National Legislature, therefore, we respectfully 
 suggest the propriety of taking into their consideration the 
 diffusion of this elementary knowledge in their several dis- 
 tricts, through the medium of the manual and chart of 
 geometry. Let the junior militia participate in the advan- 
 tages of this military science. Out of many thousands, you 
 educate one : this is well, but not sufficient. And even this 
 scanty provision is uncertain ; for the individual whom you 
 select may be less worthy of command — may be rejected ; 
 and some natural, uninstruc ted genius, from the plough, may 
 
n 
 
 supersede the cadet upon whom your exclusive patronage 
 has been lavished. Hi?ic opera et irnpensa peribunt. Ed- 
 ucate one, and neglect ten thousand, and be disappointed ! ! 
 This is patriotism inverted, tapered to a point, attenuated too 
 finely for the support of the most rarefied aerial existence. 
 No7i illud opus tenuissima stamina vincant. 
 
 General remarks on the utilitarian and tranquilizing char- 
 acter of the study of geometry, and an appeal to those loho 
 are placed in a good position to aid in its diffusion. — It 
 is desirable that the plan of the manual and chart should 
 be submitted to trial in the several districts of the United 
 States ; not where the facilities for teaching geometry are 
 already furnished, or in a class which will not advance 
 without the regular demonstrations ; but in a school, or class, 
 which will proceed to collect the facts connected with each 
 proposition, as prescribed in a preceding part of this Essay. 
 This trial cannot, however, be so extensively made, with- 
 out the kind and efiicient co-operation of the members of 
 Congress. We are therefore anxious to enli&t their influence 
 and support in behalf of this noble cause. The magnitude 
 of the enterprise fills us with regret that abler talents and 
 more propitious circumstances than those which have fallen 
 to our lot are not engaged in the execution of it. It is not 
 the high refinements of science, the deep research of philoso- 
 phy, that we would urge on the public attention; it is the 
 introduction of the principles of geometry into the schools — 
 the diflfusion of the seeds of knowledge where they are not. 
 Our object is to lay the foundation of the national educa- 
 tion in the proper and sufficient elements ; to simplify the 
 incipient means ; to scatter the good seed broadcast in town 
 and country, and send thousands of youths teeming with 
 the germes of science to seek the fostering instruction of 
 some adjacent Alma Mater; or, should that be out of their 
 reach, to ascend the hill of science by their own application. 
 
 Individual influence, however great, would be too incon- 
 siderable to warrant the expectation of any decided action 
 
in relation to this important work ; its merits will even be 
 rated by the diminutive proportions of those of our agency, 
 which only consist in a zeal for the promotion of the cause of 
 science that has increased with the progressive lapse of thirty- 
 one years. It is now our belief, based on the experience of 
 many vicissitudes, that the perpetuation of the principles laid 
 down in the Declaration of Independence depends on the 
 more equal diffusion of knowledge, the disparity of which has 
 been gradually increasing among the people ; that mathe- 
 matical science, though perhaps defective in refinement, 
 confers a greater share of power in many important partic- 
 ulars, and is therefore the best among the available means 
 of protection against any encroachment upon social rights ; 
 that the elements of this knowledge should be diffused, 
 even where they must depend solely on the congeniality 
 and vigor of the mental soil for their culture and improve- 
 ment, trusting that skill and power to combine and employ 
 them would soon appear in the native genius of the young 
 people, whom we cannot thus call up without commending 
 them to God. We are further of opinion that the studies 
 which we here desire to recommend, if carried out to their 
 legitimate conclusions, could never have connexion with 
 metaphysics, or questions of uncertain data, and consequent 
 " doubtful disputation ;" that they are wholly neutral with 
 respect to points of faith which may be controverted by the 
 various sects in polemical divinity, but unequivocally favor- 
 able in their testimony as to the goodness, wisdom, and 
 power of the Creator and Governor of the universe. They 
 are therefore incapable of being employed to promote dis- 
 order in communities, of which the inevitable consequences 
 are usurpation and oppression on the part of the rulers, un- 
 less where the laws are sufficiently powerful to restore or- 
 der to the body politic. Moreover, in the lives of distin- 
 guished mathematicians which have been handed down to 
 us, we find many able supporters of the doctrines of Christ- 
 ianity *, and there is just reason to believe that many more 
 
44 
 
 have erred, in doctrine and practice, from want of the 
 knowledge of geometry, than from its superabundance. 
 
 The study of the mathematics comes before the whole peo- 
 ple of the United States, not a thin superstratum, for accept- 
 ance or rejection. Where neither the vis inertice of the matter, 
 nor the fugitive volatiHty of the spirit of man, resists or 
 evades the consideration of the case, the want of power in 
 the agency will be sufficient to exclude the intended ad- 
 vantage. The field is very great, and the laborers must be 
 many. Who, then, will enlist with us in this cause, which 
 has for its object the greatest amount of intellectual and moral 
 good ? Will the members of the legislature refuse to con- 
 vey intelligence on this subject to the friends of popular edu- 
 cation in their several districts? May we not expect the 
 concurrence of the distinguished citizens composing the 
 membership of the National Institution in this laudable 
 enterprise? Here is a definite object presented for their 
 action, and one which lies in the entrance of the principal 
 avenue leading to the Temple of the Sciences. The means 
 here proposed require only to be submitted to the impartial 
 test of practical experiment ; the people will not forego the 
 obvious advantage of the plan. 
 
TABLE OF CONTENTS. 
 
 Page. 
 Indications of the desire to diffuse knowledge— 
 
 1. In the action of Congress with regard to the bequest of the laie Mr. 
 
 Smithson 9 
 
 9. In the establishment of the National Institution at Washington 11 
 
 3. In the action of the friends of popular education in some of the States, 
 
 resulting from the discoveries made in the last census 12 
 
 Astronomy, which leads the whole train of the sciences, dependent on 
 
 the elements of geometry < 13 
 
 A magazine of the natural sciences, for the family library, should com- 
 mence with the elements of arithmetic and geometry. . . # 15 
 
 Geometry and arithmetic should have priority of algebra 15 
 
 The magazine should explain the terms which involve principles in 
 
 arithmetic as well as in geometry 17 
 
 An explanation of the true principle of induction, as applicable to arith- 
 metic 18 
 
 The leading numbers of the magazine should not contain abstruse ques- 
 tions 19 
 
 Geometry is at war with the lawless condition of mind and matter 20 
 
 The investigations of the sciences dependent on geometry are limited 
 within the known parts of the creation ; the line and plummet must 
 
 never be out of use 21 
 
 The popular education is defective with regard to means 22 
 
 An uneducated population are the materials of a despotism., 22 
 
 Domestic education is the most available 23 
 
 Equal means of education are necessary to preserve a free government, 
 
 ami avarice is its antagonist 24 
 
 Legislation requires intellectual discipline and experience of the rela- 
 tions of society. 25 
 
 Concentration and diffusion of knowledge contrtisted 26 
 
 The metaphorical tree of knowledge 27 
 
 School commissioners and trustees are bound to lay a proper foimdation 
 
 for public instruction . . . , 28 
 
 A Chart and Manual of Geometry for the use of schools recommended 29 
 A description of the Chart and Manual of Geometry, with the title of 
 
 each, and the supposed terms of their publication 30 
 
 Geometry and arithmetic .should be taught at the same time 31 
 
46 
 
 Page 
 The efitire labor of demonstration should not be at once imposed on the 
 
 learner ; it should be classified 31 
 
 Youth is the proper lime for acquiring the elements of science, and 
 
 forming an intelligent ciiizen 32 
 
 Geometry taken in the Euclidian order is not abstruse, and maybe 
 
 learned by any good reader 32 
 
 Teachers may acquire this knowledge from the recitations of their 
 
 pupils, without any previous study 33 
 
 Colleges have a pecuniary interest in the promotion of this preparatoiy 
 
 study 34 
 
 Geometry is the law of Nature, as it regards action or motion 34 
 
 A few of the many uses of geometry to a free people enumerated 35 
 
 The study of geometry produces internal order, regulates the morals, 
 and elevates the mind to contemplate the sacred order of the Creator's 
 
 works 36 
 
 Good government is a Divine unity; and a freeman must either exer- 
 cise it upon himself, or fall into the hands of another master 37 
 
 The diagrams and propositions of geometry should be every where 
 
 exhibited 38 
 
 Geometry is necessary to complete the education of an American citizen 39 
 The General and State Governments should patronize the general dif- 
 fusion of geometry as a military science 40 
 
 General remarks on the utilitarian and tranquilizing character of the 
 study of geometry, and an appeal to those who are placed in a good 
 position to aid in its diffusion 42 
 
 NoTB. — The importance of this subject, included in the general tide, may 
 be collected, to some extent, from the several views presented in the pages 
 of this pamphlet. 
 
1 
 
 
;6AYLORDBROS. In*. 
 
 SyT«cu»», N. y. 
 
 Stockton, Caltf. 
 
 UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
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