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Yet the fact is, that these tipes are so far from being equal, that the time of descent through the arc, is less than the time of descent along the chord, in the ratio of the quadrantal arc of a circle to its diameter.
We have only to notice, farther, in reference to Mr. Keith's work, the omission of a section on geometrical analysis. This is the only branch which can be fairly applied as an instrument of investigation : and, regarded as a means of improving the mental faculties, we would rather put into the hands of youth, a series of only fifty problems or theorems, whose demonstrations should comprehend both the analysis and the synthesis, than ten times that number in whose demonstrations the analysis has no place.
These hints we present in the most friendly manner to the consideration of Mr. Keith. Notwithstanding the defects to which we have pointed, we deem his a work deserving of encouragement; and we shall rejoice to learn that he soon has opportunity to profit by our suggestions in a new edition.
Mr. Reynard's “Geometria Legitima” is a work of a very different order from the preceding. It is a treatise of bolder pretensions, but of far inferior merits. Its Author considers it as an attempt to shorten and smooth the way through the elements of geometry,' and hopes it will be found more advantageous 'to the progress of a student than the old, crooked, uneven, * and round-about Alexandrian road.' He boasts of not having 'trodden frequently in the steps of Euclid,' and' of having
made a variant course from him; expresses his astonislıment 'that geometrical subjects, intended for beginners, have never
been divided into regular and distinct heads;' (a discovery which we confess a little startled us ;) and assures us that in his work,
The theorems are all demonstrated by the direct method, which is the safest and the best way of proceeding; for to establish a truth by proving its contrary to be an untruth ay absurd suppositions, does not belong to upright geometry; it is an illiberal mode and unworthy of adoption ; (we humbly presume the Author means to say unwortky of a gentleman ;] it lessens the dignity of the subject, and is a com. plete dereliction from the direct road, which leads to the stores of science.'
Let us for awhile surrender ourselves to Mr. Reynard's guidance, and travel in his direct road.' Here we first meet with a postulate,' being, as this Author tells us, ' a self-evident truth, which is at once sanctioned by our senses, clear and inscrutable; whatever is not so, is inadmissible as a postulate.' He adds, this petition may be safely granted, as being clear, positive, and unequivocal.' If it be so at Mr. Reynard's commercial school,' we apprehend we should not be VOL. III. N. S.
very hopeful pupils; for we do not see, as yet, how that, which ' is at once sanctioned by our senses,' should be inscrutable.'
We are next told that an axiom' is å more self-evident fact than a postulate; for it gives, at once, a finite and substantial truth to the mind; clearly effected without requiring any illus-. *tration from supposition, or possibility. Then we are shown that an enunciation should comprehend only part of a proposition, and be always incomplete: then that a demonstration
proves fully to the sense the truth or falsehood of a theorem :' and soon after, that “the rolling over of a point in a straight
direction marks out the track of a straight line. By a few more such ingenious definitions and remarks, the Author's mind rolls over' in a straight line to the theorems.
Of these the general enunciation of the first two is redundant ; the words within (i. e. between) the extremities of the same are useless. Theorem 4th is not demonstrated; for the lines AB and CD might both have an inclination to FG, and yet be parallel. The 6th theorem depends upon the 5th, that upon the 4th, and the 4th depends upon the 6th. So that the Author argues in a circle' respeeting parallel lines. He also attaches a corollary to theorem the 5th, which flows from the 21st. The 8th theorem is not demonstrated, for the 4th reference is defective; the point is not established in the place referred to, but depends upon theor. 17. In theorems 14th and 15th, the triangles are not necessary alike;' and in theor. 17th, the triangle BCD is not the same as DEF, nor is it similar ; the lines are not in the same order. The corollary to theorem the 18th is not demonstrated : and in the demonstration to theorem the 23d, case the first, it is not proved that BC is equal to EF. If this be a fair specimen of Geometria Legitima, the science must have lost some of its essential characteristics since the days of Euclid.
Having thus travelled with Mr. Reynard, through what he deems demonstrations to the twenty-five theorems in the first book, we come to a series of questions to be solved. The addition of these he contemplates as a valuable peculiarity. Should he be able to consult West's Elements of Mathematics, he will find the same thing much better done; at least, he will meet with obvicus, instead of forced and unnatural examples. There is nothing, however, that we are aware of, in Mr. West's book, to compare with the following sublime and solemn passage.
• Pythagoras was so elated with joy at finding a truth so clear and so useful, (as Euc. I, 47.) and affording one of the strongest pillars of geometry, that he sacrificed to the gods a hecatomb, or one hundred oxen; thus, we have here an instance of transported zeal in the cause of learning, which shews what exquisite pleasure it must have given to this renowned philosopher, when it first appeared to his mind; and such pleasure will the young geometer continually re
ceive in his discovery of geometric truth, which will ever excel the momentary glare of pompous shews, the pursuit of inconstant fashion, or the routine of foolish pleasures ; fleeting and unreal joys are the rewards of the latter, but immortal glory and renown the boon of the former!!!
As we proceed we shall meet with other passages equally sublime. In theorem 1st, book the second, the corollary to the proposition is a part of it; and the 9th theorem is demonstrated by means of the 12th. In book the third, the 18th theorem is imperfectly demonstrated, the demonstration applying only to the case of the acute angle; the 21st and 22d con tain, each, two, and the 25th, three, distinct propositions ; the 15th of the promiscuous questions at the end of this book, demands the demonstration of a property which is not generally true; and in the 12th theorem of this book, the demonstration fails entirely. The proposition
he proposition is this :- Any two circles • which touch each other, either internally or externally, will ' have their centres and point of contact in one straight line.' They who have been accustomed to travel the round-about • Alexandrian-road, divide this proposition into its two obvious cases, and demonstrate each by a reductio ad absurdum. Not so Mr. Reynard. He goes through the matter very ingeniously, by taking the theorem for granted, in the course of his demonstration, and not being aware of it! This is the book of which the Author says, (page 80,) that he who reads it through
with steady meditation, imbibes, at the same time, such a viri‘fying principle in his mind, as will raise in him the purest Zeal, and the boldest ardour for higher speculations.'
We have no doubt it will, and are very much tempted to proceed with our Author into these “higher speculations' in the latter half of this work. But, on the whole, especially as what we have selected is a very fair sample of what follows, we think it better to relieve the dryness of these abstruse subjects, by a quotation or two from the rhetorical parts of this geometrical treatise.
Speaking of the circle, our eloquent Author breaks out into the following rapturous exclamation.
• Behold! what sublimity arises in this superior form. A form which seems to be chosen by the supreme architect of the world, in the structure of the heavens and the earth ;-it is the very basis and preservation of nature, in giving strength and durability to her constructions and omniscient operations ; the heavenly concave above us ; the wide horizon about us ; the planets revolving round the sun, and their attendants again round them, making their harmonious periods convey to our minds inexpressible delight. The appearance of the sun's daily path, strikes our senses with the most lively joy and remembrance of his constapcy and goodness, and of his support to the
nourishment of nature and existence of living creatures ; and in all God's creation it is the most beautiful of all forms; wherever it is seen to adorn, it never ceases to engage, and raise in our minds the most exquisite pleasure; therefore, for unity, simplicity, utility, and beauty, it excels all other plane figures: it is the favourite of heaver, and deserves to be divine !
O divine circle ! Once more :
. The variety of reasoning in the following book, (Book V.) as lines intersecting lines, the similarity of triangles and rectilineal figures, and their relative comparisons, when inscribed in the circle, will all sufficiently show the excellence of reasoning by proportion, the easy mode of demonstration, and the happy results arising from it; how analogies are coupled together, and a variety of conclusions consolidated into one permanent clear idea. The young geometrician will now elevate himself in the subject, a wide horizon will be presented to his view; and he will, by close and scrutable observation, be qualified to examine the most complicated diagrams, and trace the most remote relations to the very focus of the understanding.'
The preceding passages approach so nearly to perfection, in their
way, that we can only think of one possible means of improving them. About twenty years ago, a poet, whose name, unfortunately, we do not recollect, began a metrical sketch of the life of Oliver Cromwell with this line,
• Tenebrious gloom obscur'd the dismal night;'-— meaning, if we rightly interpret it,
• Dark darkness darken’d the dark dark;'Now it has struck us, that the tone of expression of this poetical genius, is so much like that of our Geometria Legitima' genius, that if he could be found and einployed in transmuting this treatise into English verse, the public would thereby be more benefited than they are likely to be if it remain in prose, however elegant, as it now stands. The minds of the British public are dull, and not easily excited to a love of the abstruse subjects into which Mr. Reynard has so profoundly dipped. We are removed only one degree from those unhappy times to which he adverts, when the mathematicians were banished the realm by a royal decree, under an accusation of their possessing the
powers of witchcraft;' is there not cause, therefore, really to tremble for him, and other men so highly gifted with this dangerous kind of knowledge, while we adopt his thrilling exclamation,
• O persecuted science! O injured reason! it seems that blind superstition, or the impious policy of priestcraft, has been a greater enemy to you than even ignorant and destructive barbarism ; the former not only confirmed prejudices against you by national yet unjust decrees, but terrified aspiring minds, and loaded genius with perpetual fetters, less to be endured than iron,' « Valuable invention
turned pale at the sight of armed bigotry, darkness was indeed spread over the earth ; and
For the rest turn to the work itself, or, as we should more conscientiously recommend, wait till the rhyming translation makes its
Art. VI. 1. Observations on Pulmonary Consumption. By Henry Her
bert Southey, M.D. 8vo. pp. 174. price 79. Longman and Co 1814. 2. Letters Addressed to his Royal Highness the Duke of Kent, on
Consumption : containing Remarks on the Efficacy of Equable and Artificial Temperature in the Treatment of that Disease. By Thomas Sutton, M.D. &c. &c. 8vo. pp. 59. price 2s. 6d Underwood, 1814. MEDICINE, it will be allowed by most persons, is already
divided into a suificient number of departments. The three separate heads of physic, surgery, and pharınacy, seem to preclude the necessity of any subordinate divisions, or more minute ramifications of the healing art. It, nevertheless, now and then happens, either from early bias, accidental impression, or some other causes, that a particular branch of one of these departments is selected by the medical artist, not for exclusive, but for prominent regard. Thus, for example, during the preparatory course of studies for the formation of a surgeon, the exquisite structure and interesting physiology of the eye shall attract, in a more than ordinary degree, the attention of the student; his reading, his researches, his dissections, and his experiments, will, in consequence, tend to a more minute and close investigation of that favourite subject; and he will come out from his studies a well instructed surgeon in general, but an oculist in respect to the feeling of particular preference. So will it sometimes happen in the pursuit and practice of medicine. The diseases of one part of the frame shall appear to deserves, in some instances, especial observation, and more than ordinary research ; and when we recollect that consumption of the langs is the giant malady of this country, that it stands first and foremost in the long list of formidable British diseases, it is not to be wondered at, that British physicians should often come out with dissertations on this most melancholy of subjects.
Within the last ten years, indeed, we have had nearly as many treatises op pulmonary consumption, all of them written by regular and respectable practitioners. To persons who are at all familiar with modern writings on medicine, the names of Beddoes, Bourne, Reid, Saunders, Buxton, Woolcombe, Duncan, will immediately occur; and to these we have now to add that which stands at the head of this article,-a name which, if it be right to make any comparison, we may be