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A LETTER 
 
 TO THE 
 
 PRESIDENT OF THE U. STATES OF AMERICA; 
 
 PROPOSING A METHOD WHEREBY THE MERITS OF 
 TEIE CONFLICTING CLAIMS OF 
 
 THE UNITED STATES AND GREAT BRITAIN, 
 ON THE DISPUTED FRONTIER, 
 
 MAV BE CORRECTLY EXAMINED AXD DETERMINED, BY SCIENTIFIC 
 
 PRINCIPLES, COMBINED WITH IND ISP LTABLE DATA, 
 
 GROUNDED IN EQUITY; 
 
 TOGETHER WITH AN EXAMINATION OF 
 
 THE TREATY OF 17S3: 
 
 FROM WHICH IS DERIVED A DEMONSTRATION OF THE FALSEHOOD OF 
 
 THE BRITISH CL\IM. 
 
 By JOHN LEE, 
 
 DECYPHEREE OF THE CHARTER OF OMAGB, AND AUTHOR OF PROPORTIONAL 
 
 FORMULA, THEORY OF LONGITODE, AND A TREATISE ON THE 
 
 ERRORS OF THE RECTANGULAR SURVEY. 
 
 ' " A}./io} S" iv ari'i&taai ri&ti voov h^qvottu itvf 
 
 Ea9'/iMV, Tou a rt noXXoi tTiavQiaxoiT^ oit-9Qw7tot, 
 Kal TE TTo/lfi; iauwae." — Iliad, Lib. XIII. 
 
 " I, wisdom, dwell with prudence, and find out knowledge of witty inven- 
 tions." — Proverbs, viii. 12. 
 
 <' Wisdom is better than weapons of war." — Ecclesiastes, ix. 18. 
 
 CAIil BRIDGE: 
 
 METCALF, TORRY, AND BALLOU. 
 
 183 9. 
 

 Entered, according to act of Congress, in the year 1839, by Metcalf, 
 ToRRY, AND Ballou, In the Clerk's Office of the District Court of the 
 District of Massachusetts. 
 
A LETTER 
 
 TO HIS EXCELLENCY THE 
 
 PRESIDENT OF THE UNITED STATES OF AMERICA. 
 
 Sir: 
 
 The conspicuous majesty of Science is now recognised 
 from nation to nation, and from clime to clime, throngli the 
 circuit of the habitable universe. From her intellectual 
 throne she swaya the destinies of men. In the remote and 
 tranquil scenes of rural cultivation; in the busy throng of 
 traffic and the ceaseless reverberating din which pervades the 
 commercial cities ; on the vast and solitary surface of the 
 undulating deep; in the barriers of eternal polar ice, or the 
 impenetrable silent horrors of subterrestrial gloom ; through 
 all the confines of the sublunary world, she claims ubiquity 
 of empire. That portion of our being, which we inherit from 
 divinity, has been, by the discipline of Science, developed in 
 such magnificence of energy as clearly marks its origin ; 
 while the petty powers of our animal constitution have sunk 
 to comparative nonentity. In the conflict of battle, and the 
 agonizing shock of war, the achievements of corporeal prow- 
 ess are behold no more; the mail-protected rank of heroes has 
 dwindled into puny insignificance, before the vast machinery 
 of destruction which Science has created and arrayed upon 
 the martial field; she impels, with invisible and superhuman 
 arm, the flame-winged missiles which annihilate legions, and 
 crush the pride of castellated bulwarks with irresistible and 
 ruinous prostration. 
 
 This development of the mental energy of man, though its 
 recent progress to maturity be characterized with gigantic and 
 
marvellous rapidity, is no precipitate result of late or sudden 
 causes. It is the gradual growth and final fructification of 
 the tree of Knowledge, through the total extent of a period 
 Mdiich has nearly comprehended six thousand years. Through 
 the reckless and barbarian atheism of lawless nnconfederaied 
 man ; through the mystic incantations and fantastic horrors 
 which mark the fearful rites of idolatrous bewildered nations; 
 and in later times, through all the seductive sophistry and 
 blasphemous impiety of audacious and perverse philosophy ; 
 like the vegetative bud which survives the changes of the 
 atmosphere, the intellectual germ has resisted every ungenial 
 influence of place and time, and preserved its undecayed vi- 
 tality, to attain the plenitude and lustre of autumnal exuber- 
 ance and bloom. 
 
 And, tlierefore, when contrasted with the flaming splendor 
 which illuminates the annals of science in the present age, 
 the page oi ancient history is not completely ^esolaie. and dark. 
 Through the mist of intervening ages, the achievements of 
 the peerless Archimedes present a monumental prototype of 
 almost every marvellous phenomenon, which, by the aid of 
 science, is created on the theatre of nations in the present day. 
 His caustic apparatus and projectile engines refer the imagin- 
 ation, by a simultaneous and triple analogy, to the sudden 
 flash and heavy roar of the cannon, to the lightning and 
 thunder of the electric machine, and the irresistible energy 
 which results from the concentration of the optic ray. His 
 investigations in the theory of motion, equilibrium, and im- 
 pulse, have served as- a basis for all that we now know in 
 mechanical science, through all its modifications and depart- 
 ments, comprehending the multitudinous motive powers and 
 impulsive agencies of solid and fluid materials. And in bright 
 and glorious addition to all the preceding discoveries, his dar- 
 ing and sublime excursions, in the intellectual region of pure 
 Geometry, through unexplored and solitary tracts of knowl- 
 edge, but copious and luxuriant in the intrinsic and immortal 
 beauty of the immutable nature of things ; these intrepid 
 fiio'hts of reason have directed the research of every subse- 
 quent adventurer on that boundless and eternal field. 
 
Sir, it is not necessary here, to enumerate, in detail, the im- 
 portant services for wliich celebrity is due to otlier numerous 
 and illustrious names, which adorn the recorded catalogue of 
 scientific labors and lore, through the long and shadowy tract 
 of time comprehended in ancient history ; though the mental 
 faculties of your Excellency must have been, for years, indis- 
 pensably pre-occupied with all the responsibilities and cares 
 attached to the superintendence of the political alfairs of a great 
 and rising nation; I consider it yet a presumable circumstance, 
 that all the classic recollections, which refer to the develop- 
 ment and progress of the human mind from early imbecility 
 to intellectual manhood, have not completely faded from your 
 memory. 
 
 Though the extraordinary and startling problem, the per- 
 formance of which is the professed and immediate object of 
 the present letter, be one which is indeed unexampled in the 
 history of science, the preceding rapid sketch of the continu- 
 ous movements and incessant researches of the human mind, 
 will, I am disposed to believe, diminish the deep surprise, 
 with which, for the first time, your Excellency, and the public 
 wouldotherwise receive the intimation of theattcmpted solulion 
 of this problem. By a simple and direct pursuit of the ideal 
 train of associations presented in the preceding sketch, the 
 solution of this problem will easily apj)ear to he part and par- 
 cel of the connected results of the unquiet and aspiring ope- 
 rations of the general human mind ; of that primeval, 
 celestial, and elastic impulse, whereby our universal race and 
 lineage are exalted above the humility of their native dust, to 
 tread, in intellectual pre-eminence, the ulterior confines of the 
 vaulted universe, and contribute recognition and homage to 
 the invisible Architect who dwells beyond. 
 
 Sir, I am a native and voluntary subject of the British 
 sovereign ; but iew and doubtful are the claims of gratitude, 
 however, which the British nation, or any other, has on my 
 individual affections; this country I have sought, not as a 
 refuge for political disaffection, or a laboratory for the experi- 
 mental career of a political empiric; not as the ultimate re- 
 source of one who escapes from the horrors of penury, nor 
 
yet as a protecting asylum for an infamous and guilty fugitive ; 
 I have directed my steps upon the transatlantic shore, to 
 obtain a refuge, in retirement, from scenes of adversity, and 
 a cessation of the miseries of mental distress ; to seek a rest- 
 ing place where sources of private unhappiness may feel the 
 balm of solitude ; where the eyes of the depraved rabble may 
 never enjoy that luxury which is deemed intensely grateful to 
 malignity and envy, when permitted to espy the faded lustre 
 of decayed respectability, and the ruinous operations of disas- 
 trous and reverted fortune ; where the afflicted proud man, 
 when he feels the burning and resistless impulse to relieve and 
 vent his anguish by external or internal tears, may weep in 
 secret and alone. 
 
 But while the self-dependent and secluded nature of my 
 present condition compels me to regard my fealty as my own, 
 and brook no infringement of my personal rights from any 
 existing human power, when the means of retaliation are ac- 
 cessible ; I cling to the belief, that I feel myself too deeply 
 bound by the obligations of moral conscience, by the intrinsic 
 sanctity of justice, by the love of equity which dwells immu- 
 tably inherent in the virtuous mind, and by the lingering at- 
 tachments of pre-existing national sympathy which distinguish 
 and adorn the human character, and interpose a true criterion 
 betwixt the magnanimous and the servile spirit ; I believe 
 myself too potently restricted by these considerations, to es- 
 pouse the American claims for the sake of popular favor, 
 in considering the merits of the great international dispute 
 which I propose to examine in this letter. 
 
 I must also observe to your Excellency, that, while I con- 
 ceive myself, in this country, deeply indebted to a few indi- 
 vidual friends; yet, from the nation at large, or any section, 
 great or small, of the community, I derive no favor, and there- 
 fore acknowledge no obligation ; I anticipate no advantage, and 
 seek no benefit, except as the reward of my own exertions; 
 and few therefore, I presume, can be easily found, who are 
 better qualified by total exemption from ''-fear, favor, or affec- 
 tion,'' than I am, for the examination of this great question. 
 
 Having offered the preceding remarks to your Excellency, 
 
for the vindication of my own character, conduct, and motives, 
 m attempting to determine the merits and dispel the perplexi- 
 ties of this dispute, I now proceed, without further preface 
 or delay, to enter on the field of explanation and decision. 
 And here, in the audible and free communication, and the 
 open conspicuous prospect of the great and general amphithea- 
 tre of nations, I avow myself constrained in justice, to give 
 my direct and unequivocal declaration and opinion on the 
 side of the American party in this litigation. I am constrained 
 to acknowledge, that every feature of the British claim so 
 visibly and palpably betrays the stamp of artificial and recent 
 origin, as to seal the lips of every conscientious and enlighten- 
 ed man who attempts to speak in its deience. I am constrain- 
 ed to deprecate, with horror and disdain, the dark, the damna- 
 ble, perverse, and baleful doctrine of misguided ethical 
 philosophy, which tolerates an act of iniquity and fraud, if 
 such act be the act of a nation ; which designates extortion by 
 the title of refined policy, if a nation, as one man, have deter- 
 mined to act in concert as unanimous confederate extortioners ; 
 which covers political villainy with gaudy epithets, and hides 
 the infamy and shame of a people in the frivolous delusions of 
 insignificant verbal vanity ; which encourages the national 
 and public perpetration of deeds, the doom and consequence 
 of which, to the performer, if performed by one individual, 
 would eject him, as a vile, abhorred, and solitary fugitive, 
 abroad upon the face of the earth. I am constrained to ac- 
 knowledge, that if the lion be correctly taken for a symbol of 
 the nobler characteristics of the British nation, the same anal- 
 ogy also too unhappily and closely prevails, in respect of some 
 other less attractive traits of character which mark the royal 
 quadruped ; I am constrained to express my fear, that, if con- 
 sidered in reference to such a comparison, the acts of that 
 nation have more than once betrayed a propensity — royal in 
 cupidity, and more than royal in rapacity. 
 
 In the examination of the question of the frontier, which I 
 propose to undertake, on scientific principles, in the subsequent 
 part of this letter, many mathematical theorems are involved, 
 of so technical a nature, that I should be guilty of gross and 
 
contemptible hypocrisy, by pretending to presume that all 
 those theorems are familiar to your Excellency, exercised and 
 occupied as you have been for many years, in the responsibili- 
 ties and duties of political affairs. For the satisfaction of 
 your Excellency concerning those technical inquiries, I pre- 
 sume to suggest a reference, if necessary, to the mathematical 
 authorities of Yale, Virginia, and West Point. 
 
 Finally, Sir, before I undertake the proposed examination, 
 I must express the hope, that a consideration of the nature of 
 my attempt will tend, with additional force, to impress upon 
 your Excellency, a deep conviction of the incalculable benefits 
 resulting to maiikind, in every conceivable department of 
 human affairs, from the patronage, diffusion, and protection of 
 science. I shall rejoice, if enabled hereafter to believe, that, by 
 any observation contained in or suggested by this letter, the 
 attention of your Excellency, or that of any other influential 
 man, has been more closely directed to the promotion of that 
 glorious object. 
 
 But I cannot incur the deep culpability of neglecting to 
 inform your Excellency, and all my fellow-men, that mathe- 
 matical science, like every other human pursuit, has its vani- 
 ties, deceits, and snares. More especially in later times, a 
 race of men has appeared upon the field of demonstration, 
 the devious meteoric aspect of whose intellectual career has 
 induced me to apply to such indviduals the epithet, Formu- 
 larian, as a distinctive afjpellation. These men shun defini- 
 tions, and refuse to explore first principles ; they conceal am- 
 biguity by apparently precise, but latently equivocal words; 
 they accomplish, on formulcp, unusual and curious transform- 
 ations, by unjustly, though plausibly, gcnercdizing the 
 ordinary rides of opei^ation, like certain other philosophers, 
 "who, having once discovered that a circumstance is very 
 generally true, immediately arm themselves with such a 
 discovery, as a weapon wherewith to deiiy the reality of every 
 case of exception which may afterwards occur. The absurdi- 
 ties of these men are sometimes harmless and amusing ; but 
 unhappily, too often, more calamitous results ensue. Among 
 this visionary tribe, La Place appears to be pre-eminent, in 
 
propensity to mischief, and in power of performance. The 
 splendid fictions and marvellous delusions of that unrivalled 
 Formidarian are pregnant with destruction ; they are sub- 
 versive, in their final tendency, of religion, morality, and 
 social order; and consequently hostile to the present and sub- 
 sequent happiness of men. Such individuals may be rightly 
 regarded as the despicable vermin which pollute, while they 
 prey upon, the intellectual decayed remains of Archimedes 
 and of Newton. 
 
 Before I undertake the proposed question, I shall finally 
 observe to your Excellency, and the public, that, concerning 
 the promotion of science, I am sorry to perceive the existence 
 of an error, which, in politics and literature, is equally fatal 
 and disastrous. This error consists in the supposed utiliti/ of 
 domineering behavior. One of the predecessors of your 
 Excellency, in the exalted station which you now occupy, 
 unfortunately tarnished, by a fatal error of that nature, the 
 lustre of his previous reputation ; but how much more inde- 
 scribably contemptible is the aspect of a domineering bully 
 among juvenile students^ than that of a rash and arbitrary 
 man directing the affairs of a nation ! The only considera- 
 tion, that operates with resistless and perpetual sway on every 
 department and rank of human society, is the prepossessing 
 native dignity icldch peculiarly discriminates the gentleman ; 
 and which is completely incompatible with every vestige of 
 that repulsive rudeness, which betrays the barbarian, the 
 ruffian, or the bear. 
 
 Finally, I now proceed, according to proposal, to furnish 
 in detail, to your Excellency, the scientific investigation and 
 solution of the problem respecting the disputed frontier. 
 
10 
 
 INVESTIGATION OF THE PROBLEM OF THE DISPUTED 
 
 FRONTIER. 
 
 Art. 1. If two parties agree to discriminate a tract of 
 country into two several shares, by a transverse boundary, and 
 the face of the country present no natural obstacles, that 
 boundary will, beyond all doubt, be Rectilineal; for every 
 curvilinear deflection or angular deviation would be a source 
 of improfitable toil and useless perplexity. 
 
 Art. 2. And hence if ixny part of such boundary were 
 disputed or defaced, it might be, at any time, re-ascertained, 
 by simply pursuing the direction of that part which is known. 
 
 Art. 3. But if natural obstacles occur on the face of the 
 country, they will operate in two ways ; first, by intercept- 
 ing the course and precluding the progress of the intended 
 rectilineal boundary ; and secondly, by presenting, in them- 
 selves, a range of immutable and definite local features, which 
 may, with facility and certainty, be taken as connecting-points 
 through which we may imagine a boundary to pass. 
 
 Art. 4. Between the parties, however, a previous under- 
 standing must have existed, either strict, precise, and immu- 
 table, or else restrained within certain limits of allowable 
 adjustment, concerning the proportionate magnitudes and 
 relative situations of the two shares. 
 
 Art. 5. And hence, in the selection of such natural con- 
 necting-points, to mark the direction of a boundary, the pro- 
 ceedings will be governed and affected by this indispensable 
 consideration ; so to direct such a boundary, as to violate in the 
 smallest possible degree, the understanding which existed, 
 respecting tlie situations and magnitudes of the two shares. 
 
 Art. 6. And if the series of natural points discontinue, 
 before this boundary has completed the extent of its course, 
 and the said boundary, having reached the last of these points, 
 emerge upon the uniform face of the country, so that, beyond 
 that point, its direction be optional; the only consideration 
 thenceforth existing, which can regulate the course of that 
 final part of the boundary, is this : so to direct this final part, 
 
11 
 
 as to compensate for any violation of the aforesaid previous 
 ujiderstanding, which may, of necessity, have arisen, by adopt- 
 ing- this train of natural points, to regidate the course of the 
 preceding part. 
 
 Art. 7. And this consideration is equally important and 
 decisive, at either extremity of the said boundary, or at both ; 
 the term ^^final,'^ in the previous article, being merely adopted 
 for facility of expression ; since, of the two extremities of any 
 such boundary, either may be taken as the initial, and the 
 other as the final extremity. 
 
 Art. 8. The import of Articles 6, 6, and 7, connectively 
 taken, is expressible thus. In the prescription of any such 
 boundary, the total course of the proceedings will be govern- 
 ed by this consideration ; .so to accommodate the several parts 
 of that boundary to each other, as to give to the total boundary, 
 precisely, or as nearly as possible, a self-adjusting character ; 
 that is, that all infringements of the general prc-under stood 
 conditions, which are unavoidably made in certain parts of 
 that boundary, in favor of one party, are counteracted and 
 recompensed by other infringements which are elsewhere made 
 in the course of that boundary, in favor of the other party. 
 
 Art. 9. Now, as we have observed in Article 1, that all 
 ])oundaries, if not intercepted by obstacles, would be rectilin- 
 eal ; and as all desirable conditions, concerning the relative 
 situations and magnitudes of two shares, can be satisfied by 
 merely accommodating the situation and direction of a rec- 
 tilineal boundary to each particular case; it fdllov/s, that, 
 whenever the separation and distinction of two several shares 
 is required, a straight line might be drawn somewhere, which 
 would satisfy the general pre-understood conditions between 
 the parties. Such a straight line we shall call a Normal. 
 
 Art. 10. From Arts. 8 and 9 we learn, that if any bounda- 
 ry whatever between two shares be completely self-adjusted, 
 the following remarkable relation must be verified between it, 
 and evej-y normal ichatever, appertaining to the same case 
 between the same two parties. In Fig. 1, let N n be a nor- 
 mal, and ABCDEFG be a self-adjusted boundary, Q q and 
 R r being Qxterior boundaries of the total tract which com- 
 
12 
 
 prehends the two shares ; then, the sum of all the areas, 
 a, (i, y, s, &c., which are mtercepted by A^ n, and those parts 
 of the tortuous boundary which lie on one side of it, is equal 
 to the sum of a, b, c, d, &c., which are intercepted by A'' n, 
 and by those parts of the tortuous line which lie on the other 
 side of it. For, if the two sums be unequal, the straight and 
 the tortuous boundary cannot both satisfy the understood con- 
 ditions of the relative magnitudes of the two shares ; since, 
 by the adoption of one boundary, one party has a greater 
 share, and tiie other a less, than by the adoption of the other 
 boundary ; but both boundaries do satisfy the aforesaid condi- 
 tions ; A^ w, by its character as a noi^mal, and the tortuous 
 boundary by self-adjustment ; therefore the two sums must 
 be equal. Q, . E . D. 
 
 Art. 11. Whenever a straight line bears to any other line 
 such a relation, with respect to the intercepted areas, as that 
 which N n bears to the tortuous boundary in Art. 10, we 
 shall call that straight line, a coequator to the other line. 
 
 Art. 12. But yet, among tortuous lines, the general bear- 
 ing or progressive tendency of one, may depart from that of 
 its coequator, much more than that of another tortuous line 
 from its coequator. Thus, in Figs. 2 and 3, the general bear- 
 ings of the tortuous lines ABCDEF, and a b c d e f, exactly 
 coincide in direction with their coequators ^F and af, Avhile 
 the general bearing of a b c d e fg h, in Fig. 4, is itself the 
 tortuous line a d i, intersecting the coequator a 7c, AB and 
 CD being the exterior boundaries comprehending the total 
 tract which contains the two shares. Whenever the coequa- 
 tor and the general bearing exactly coincide, the two extremi- 
 ties of the coequator and those of the tortuous line will also 
 exactly coincide, as evidently appears by Figs, 2 and 3. This 
 relation we shall express by saying, that, in every such case, 
 the tortuous line preserves perfect affinity with its coequator ; 
 but in other cases, the want of such affinity shall be express- 
 ed by stating, that it more or less diverges from that coequator. 
 
 Art. 13. Before proceeding further, it is well to consider, 
 in what circumstance this greater or less divergency cojisists. 
 Two straight lines, which coincide in direction, make no an- 
 
13 
 
 gle. Now in Figs. 2 and 3, the coequator AF, in the former 
 case, and a f, in the latter, coincides, or is identical with, a 
 straight line uniting the extremities of the boundary to which it 
 is draivn as a coequator. In each case, the coequator makes no 
 angle with the connecting Hne, and therefore we say that such 
 boundary has no divergence from that coequator, or has perfect 
 affinity with it, by which we mean the same thing. But, in 
 Fig. 4, the straight line a h, which connects the extremities 
 of the dotted boundary, does make an angle with the coequa- 
 tor a Jc, and therefore we say that such boundary diverges 
 from that coequator. Lastly, in Fig. 5, if a b c d he a bounda- 
 ry, and a d, ef and ^'^ h, be all coequators, AB and CD being 
 the exterior boundaries, comprehending the total tract which 
 contains the two shares; we perceive that abed has no di- 
 vergence from a d, but it has a divergence from ef, as we per- 
 ceive by the angles " and (f, which ef makes with a d. But 
 abed has a yet greater divergence from g h ; for g h makes 
 angles with a d, which severally exceed the two former, by 
 the angles '/ and •'■ 
 
 Art. 14. But here an indispensable remark must be made. 
 For, since any two straight lines, which meet, will make, if 
 produced, as in Fig. Q, four a7iglcs equal by pairs, as a and «, b 
 and 1^' and since, if either pair be very small, the divergence of 
 the two lilies will be very small ; therefore, of two adjacent 
 angles, which one straight line makes with another, as the 
 two angles a and b, in Fig 7 ; if these two be unequal, the 
 minor angle is that whereby we measure the divergence. 
 
 AiiT. 15. Now as it appears from Arts. I and 9, that all 
 boundaries, if not encountered by natural impediments, would 
 be Norrnals, because boundaries of such a character offer to the 
 human understanding the only obvious, primary, and imme- 
 diate resource whereby to realize the pre-conceived conditions 
 between the parties ; it follows, that where a boundary has 
 been eventually adopted between two such parties, the frst 
 notion of such boundary must have been that of a noimal^ 
 which, for the sake of accommodation to existing circum- 
 stances, has been diversified and altered into that form under 
 which it was finally adopted. In every such case, the pre- 
 
14 
 
 conceived original normal we shall call the Archetype, and the 
 actual boundary therefrom derived we shall call the Metatype. 
 
 Art. 16. Hence we obtain a test for the detection of a spu- 
 rious boundary. For if we can collect any evidence to show 
 that such boundary has been obtained and derived from no 
 normal archetype, we have established a proof, that such boini- 
 dary was never adopted as the result of the mutual delibera- 
 tion of two contracting parties, but betrays a fictitious origin 
 by the devious and distorted aspect of its general career ; pre- 
 senting, indeed, no unsuitable picture of the inconsistent, 
 perverse, and guilty movements of the creating spirit which 
 directed it ; alternately impelled and restricted by the stimu- 
 lations of encroaching avarice, and the retiring trepidation of 
 abashed and conscious turpitude. 
 
 Art. 17. To all that feel disposed, without reflection, to 
 denounce the principle detailed in the preceding Article, as 
 far-fetched and fanciful, it may be well to observe, that «/" that 
 principle be a mere metaphysical and idle conceit, the fact is 
 very extraordinary, that, from the most primitive and unspec- 
 ulating ages of the human race, down to the present time, the 
 universal structure of language will clearly prove, that, from 
 the geometrical ideas of " Siraight,^^ and ^^ Crooked,^' ^'Even," 
 and " Uneven,^^ the moral ideas of ^^Good,''^ and "^i;«7," have, 
 in virtue of the right of kindred, borrowed their verbal habil- 
 iments. With moderate research, and possibly with some sur- 
 prise, those objectors will discover, that ^'Equity,'' is ''Flat- 
 ness,^' and moral Mectitude is moral Straightness. They will 
 also perceive that a ''Delinquent^' is one who siverves from 
 the path of duty, so as to " Leave"" that path lying ''Off" or 
 away from his present course. They will also further perceive, 
 that a "Perverse" disposition is that which is "Very Much 
 Twisted Away." To multi})ly examples of this kind is not 
 necessary ; they are familiar to every linguist, and clearly in- 
 dicate an intrinsic similarity and a usual association of these 
 geometrical and moral ideas, for which no satisfactory reason 
 can be easily given, if we deny the proposition, that a tendency 
 to straightness will be a prevailing feature in a boundary 
 traced for the first time between two tracts of land, under the 
 direction of honest men. 
 
Art. 18. The process of deriving a boundary irom its nor- 
 mal archetype will be evidently guided and ruled by this con- 
 sideration ; to shun mis-regulation, perplexity, and toil, hy di- 
 recting this boundary, so as, first, to render it self-adjusted, 
 whereby the normal archetype will be its coequator ; a7id sc- 
 condly, so as, if possible, to have perfect affinity with its nor- 
 mal archetype, or else to diverge from that archetype in the 
 smallest possible degree. A familiar illustration of this case 
 may be drawn from that of a traveller, who, being led by an- 
 other person through an unknown country, will deviate from 
 the trade of his guide as little as possible. 
 
 Art. 19. Hence we obtain a satisfactory test whereby we 
 may discover, at least in extreme cases, whether any boundary 
 has been derived from a normal archetype, or fabricated on some 
 other principles. For if we can discover that such boundary 
 diverges extravagantly from each of all its possible coequators, 
 wc have clearly a strong presumption that such boundary ivas 
 not obtained from any normal archetype ; and, on the other hand, 
 if we discover that such boundary has 07ie coequator to which 
 it has perfect affinity, or from which it very slightly diverges, we 
 have an equally valid presumption of the opposite kind. 
 
 Art. 20. But a question may occur, of the same general na- 
 ture, but under a diilerent form, wherein the preceding test 
 may be safely applied, not only in extreme cases, but in any 
 case. For if two boundaries be placed, from the peculiar 
 cause, which gave them origin, or from any other circum- 
 stance, under such conflicting conditions and relations, one to 
 the other, that one of these boundaries 7nust have been derived 
 from a normal archetype, and the other must have been fabricated 
 on other principles ; and if we ascertain that we can draw 
 to one of these boundaries a coequator to which it has perfect 
 affinity, whereas none such can be drawn to the other; or else, 
 if we draw to each boundary that coequator from which it 
 has least divergence ; and then discover the divergence of the 
 former boundary from such coequator to be less than that of 
 the latter from its own coequator ; we obtain thus, for the for- 
 m.er boundary, a cause of preference before the latter, which de- 
 cides the question. 
 
16 
 
 Art. 21. It is also observable that sometimes a coequator 
 may be parallel to the straight Hne which connects the ex- 
 tremities of the boundary, as de to ac, which connects the ex- 
 tremities of the boundary ah c^ in Fig, 8 ; or, though not par- 
 allel, may not meet that straight line, unless both be produced 
 beyond the total tract, as the coequator/^, in Fig. 9, meets a e 
 in g, the exterior borders of the tract being ABCDE and 
 I GHI. All such coequators we shall designate, for the sake 
 of distinction, as Remote, and all others as Adjacent. From the 
 extreme difficulty, however, of directing an adjusted boundary 
 by any such remote coequator, it is needless to say that no 
 boundary has ever been derived from such a coequator as its 
 normal archetype. In the examination, therefore, of the com- 
 parative claims of difierent boundaries, to a genuine origin 
 from a normal archetype, we have no cause to make any in- 
 quiry with respect to remote coequators. 
 
 Art. 22. If two boundaries be so circumstanced, that ojie 
 must be spurious and the other genuine, but — which to ac- 
 knowledge as genimie, — and which to reject — we are uncer- 
 tain ; the only necessary aid, which has not yet been furnished 
 for the settlement of such a question, is a process whereby we 
 can try the question of greater or less divergency, as stated in 
 the preceding Articles. For, by such a process, we can pre- 
 sumptively ascertain, as appears by those Articles, which of 
 those boundaries has been derived from a normal archetype, 
 and which was fabricated on other principles ; and again by 
 Art. 16, from the discovery of these latter facts, we are imme- 
 diately enabled to discriminate the spurious from the gen- 
 uine boundary. 
 
 Art. 23. From each of the two extremities of any bounda- 
 ry a coequator to that boundary may be drawn. For, let a?iy 
 straight line, as dein Fig. 10 and 11, be drawn from one ex; 
 tremity, as d, of a boundary abed, till it meets the opposite 
 exterior border of the tract in e; de either not again meeting 
 the boundary, as in Fig. 10, or again meeting it one or more 
 times, as in Fig. 11. In the former case, let the area of the 
 space comprehended between the boundary abed, the straight 
 line d e, and the border a e, be calculated and represented by 
 
17 
 
 A ; it is now an easy geodesic problem, to determine the posi- 
 tion of a straight line, as the doited line da, which will make 
 the space comprehended between itself, de, and the border, 
 equal to A. The straight line d a is the required coequator. 
 For since d c augments one of the shares, as determined by the 
 boundary, by the quantity A ; and again d a diminishes by the 
 same quantity A, the share which has thus been augmented ; 
 that share, and consequently the opposite one also, is now re- 
 stored to its former value ; that is, da is a coequator. In the 
 latter case, let all the areas intercepted by d e, by the bounda- 
 ry, and by the opposite border, on one side of de, be collected 
 into one sum ; and let all such areas on the o^/tcr side of de he 
 collected and summed in the same manner ; and if, on one 
 side, no intercepted area be found, the sum of such areas on 
 that side may be stated as equal to zero ; take the difference of 
 the two sums, and denote it by t^; then, upon that side of de, 
 where the sum of the intercepted areas is greater than the 
 other, draw df so as to make the space comprehended be- 
 tween itself, de, and the border, equal to (5; then for the same 
 reasons which were assigned in the preceding case, dfmust 
 be a coequator. Every coequator, which meets one of the ex- 
 tremities of a boundary, shall hereafter, for the sake of distinc- 
 tion, be styled Conterminal, and every other coequator, Dis- 
 terminal. 
 
 Art. 24. All coequators to the same boundary must meet 
 and intersect within the exterior borders of the total tract 
 which contains the two shares. For if two straight lines meet 
 on such exterior border, as a c and b c. Fig. 12, which meet on 
 the border CD ; or if they meet neither on nor within such bor- 
 der, as a c and h d. Fig. 13 ; then one of the two shares, deter- 
 mined by one of these straight lines, is augmented by the other 
 straight line, without any counteracting diminution ; therefore 
 they cannot be both coequators. Therefore any two coequa- 
 tors must intersect within the borders, as a i and c d, Fig. 14. 
 
 Art. 25. And hence it immediately results, that, from either 
 extremity of a boundary, we can draw but one coequator; 
 3 
 
18 
 
 tiiat IS, to cyeiy boundary, we can draw tioo, and only tw 
 conterminal coequators ; and, in the case of perfect affinity, these 
 two coincide in direction, and thus, in a certain sense, become 
 one. 
 
 Art. 26. Of all coequators drawn to the same boundary, 
 remote ones excepted, one of the two conterminal coequators 
 is that which has least divergence. For, in Fig. 15, let a and 
 b be the two extremities of a boundary, which it is not neces- 
 sary here to represent ; let a c and b d be the two conterminal 
 coequators, intersecting in g ; let e Tc be any other coequator 
 which is not remote, intersecting a c in h, and b d in i ; be- 
 cause e it is not remote, it meets the connecting line a b, at 
 some point/, between a and b; now of the triangle i/ i, the 
 exterior angle « is greater than the angle at b; and of the tri- 
 angle afhthe ^exterior angle ^ is greater than the angle at a ; 
 therefore, of « and /?, the minor one, if these be unequal, or 
 each of them, if equal, is greater than one of the two interior 
 angles at a and b ; that is, in any case ; one of the two interior 
 angles at a and b, is less than each of the angles at « and /s. 
 Of that interior angle and its adjacent exterior, if the interior 
 be the minor angle, or if both be equal, the conterminal coe- 
 quator, to which that angle appertains, must have less diver- 
 gence than e h: ; but if the adjacent exterior angle be the mi- 
 nor, then, since the interior angle is less than each of the two 
 angles « and ,-?, a fortiori, the exterior angle, which is now the 
 minor, is less than each of the angles a and P ; therefore, in 
 every possible case, one of the two conterminal coequators has 
 less divergence than e Tc. Q. E. D. 
 
 Art, 27. Hence, if the conflicting claims of two boundaries 
 be such, that one must be spurious and the other genuine, we 
 have the following practical rule for a decision -oi the case. 
 To one boundary, draw both its conterminal coequators, and ascer- 
 tain the divergence of each, selecting the lesser divergence, or ei- 
 ther, if equal, as the least possible which that boundary can have 
 with any coequator which is not remote ; in like manner, discover 
 the least divergence, in the case of the other boundary; compare, 
 these two results together ; then, whichever boundary affords the. 
 
19 
 
 less result, we obtain, by the principles detailed in the preceding 
 articles, a preponderating evidence in favor of that boundary. 
 
 Art. 28. By the mere mental substitution of arcs of great 
 circles for straight lines, the whole preceding theory, with 
 scarcely a verbal alteration, becomes directly applicable to all 
 cases, in which, from the large extent of the total tract, we 
 may suppose the rotundity of the terrestrial surface to have 
 any perceptible effect on the question. 
 
 Art. 29. But, in the practical application of this theory, 
 the following indispensable precaution must be observed. 
 Through all the preceding investigations, the extent of any 
 one boundary is conceived to be that wherein it separates, 
 every where, the previovsly undetermined shares of those two par- 
 ties, and of those two parties alone. And therefore, if a bounda- 
 ry be part of a continuous frontier ; and if that frontier, in 
 other parts of its course, be a boundary to shares predetermin- 
 ed between the same two parties, or a boundary at all apper- 
 taining to any share of any third party ; if we would apply 
 the preceding investigation to this first boundary, we must 
 first ascertain hoio mucli of that frontier constitutes this boun- 
 dary, by discovering on what points of that frontier the extrem- 
 ities of this boundary fall. 
 
 Art. 30. Hence, in the application of the preceding process 
 to an examination of the two boundaries respectively claimed 
 by the American and British nations, on the frontiers of New 
 Brunswick and Maine ; we should first ascertain where the 
 common ivestward extremity of the tico covficting boundaries 
 lies ; that is, to what extent westward, in 1783, the boundary 
 established in that year separated British from American ter- 
 ritory. The true answer to this question seems to be, that 
 such boundary was bona fide a boundary between the United 
 States and the British possessions, through all its westward 
 course till it first reached a branch of the Mississippi ; as every 
 part of the region westward of that river, at least as far as to 
 the Rocky Mountains, appears to have been, at that time, ei- 
 ther Indian or French territory. This question, however, be- 
 ing subject to the decision of historical research, its further 
 consideration is unnecessary here. 
 
20 
 
 AN EXAMINATION OF THE TREATY OF 1783 : 
 
 FROM WHICH IS DERIVED 
 
 A DEMONSTRATION OF THE FALSEHOOD OF 
 THE BRITISH CLAIM, 
 
 AND ALSO OF 
 
 THE TRUTH OF THE AMERICAN. 
 
 Art. 1. An angle of any superficial figure, lies at a point 
 where two sides of the figure meet, as the point A in Fig. 16. 
 
 Art. 2. Two sides of any such figure can meet only at their 
 extremities. 
 
 Art, 3. Hence the angles of any such figure can exist only 
 at the extremities of the several sides. 
 
 Art. 4. Hence, any such figure has no angle at a point on 
 any of its own sides, between the extremities of that side. 
 
 Art. 5. It may easily, however, have an angle of another 
 figure, or several angles of as many several figures, at such a 
 point on its own side, between the extremities of that side ; as 
 the angles a and c, at the point Z», in Fig. 17, where AB, BC, 
 CD, and DE, are sides of a figure whereof the total repre- 
 sentation is not necessary here ; and h d is u. common side of 
 two other figures, whose representation is also unnecessary. 
 
 Art. 6. Before, during, and after the preparation of the 
 treaty of 1783, for a certain space of time, the two tracts of 
 country, which are now called Nova Scotia and New Bruns- 
 wick, were comprehended under the common appellation of 
 
21 
 
 Nova Scotia. See the article, ^' Nova Scotia,'' in the En- 
 cyclopedia Americana, and also in Dr. Rees's Cyclopedia. 
 
 Art. 7. In that comprehensive sense, we shall employ the 
 term in this examination. 
 
 Art. 8. Now it appears from the map of Lower Canada, 
 with adjacent parts of the United States and Nova Scotia, pub- 
 lished by the British " Society for the Diffusion of Useful 
 Knowledge," that ojie side of Nova Scotia, namely, the west- 
 ern, is a line extending North and South, as BC in Fig. 17 ; 
 whose northern extremity C meets another side, namely, CD, 
 and its southern extremity meets the Chiputnaticook river, 
 which, by the line of its progress to Passamaquoddy Bay, 
 makes a third side. It also appears from that map, that, ac- 
 cording to the British claim, the " northwest angle of Nova 
 Scotia,'' specified in the treaty of 1783, lies on the point b, 
 of the side BC, between the extremities of that side, and b d 
 is a common side or boundary of the United States and of 
 Canada. But, by what we have shown in articles 4 and 5, 
 the angle a would, in that case, be the southeast angle of Can- 
 ada, and the angle c would be the northeast angle of the Uni- 
 ted States ; but Nova Scotia has no angle whatever at the 
 'point b ; therefore the British claim must be false. 
 
 Art. 9. But, by the American claim, the northwest angle 
 of Nova Scotia lies at the point C, where undoubtedly there 
 is an angle of Nova Scotia; which angle, being also the only 
 angle of Nova Scotia which is made by the northern extrenii- 
 ty of the side BC; ii must be the northwest angle of Nova 
 Scotia, specified in the treaty aforesaid. Therefore, the Brit- 
 ish claim is false, and also the American is true. Q. 
 E. D. 
 
 I cannot conclude the foregoing investigations, without ac- 
 knowledging my obligations to Dr. J. D. Hedge, of Cam- 
 bridge, for the kind and active manner in which he has labored 
 to ensure the correctness, encourage the publication, and pro- 
 mote the success of this work, by the removal of many prac- 
 
22 
 
 tical difficulties, which, otherwise, if not insurmountable, 
 would have created extreme delay and embarrassment ; more 
 especially I refer to the exertions of that gentleman for the 
 facilitation of my access to authorities and documents, which 
 I should otherwise have _found it extremely difficult, if not 
 impossible, to procure. 
 
 Finally, Sir, having now completed a task, in the perform- 
 ance of which I have been animated only by an impulse of 
 duty, combined with an encouraging hope of inheriting a share 
 of that immortal celebrity, which the man who benefits his 
 race by intellectual achievements may extort from the grasp 
 of an unwilling world, and retain to the latest extent of hu- 
 man posterity ; 
 
 I subscribe myself, 
 
 with great respect, 
 
 your Excellency's 
 obedient servant, 
 
 JOHN LEE. 
 
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