r Let AD and BD be two tangents to the points A and B, drawn from the extremity of the chord AB, the point where they intersect at D will make AD equal BD. From A and B draw AC and BC to the centre of the circle, then we have shown (Theorem v. Part II) that the angles DAC and DBC are both right angles. Now, as AC equals BC, the angles ABC and BAC (by Theorem iv. Part 1.) are equal; now if we take these equal angles from the angles DAC and DBC, the remainders of those angles, viz DAB and DBA will also be equal to each other; hence, in the triangle DAB, the angles at A and B being equal, the sides AD and BD must also be equal (by Theorem iv Part 1.) Hence, the tangent AD and BD are shown to be equal, as was required. the chord corresponds; that is, supposing the chord to include 60 degrees of the whole circumference, the angles the tangents make with the chord will be each 30 degrees, and the angle the two tangents make with each other will be 120 degrees, or a semicircle (180 degrees) wanting 60 degrees. COROLLARY II.-Hence, if the tangents are perpendicular to each other, the chord from which they are drawn is a chord of 90 degrees, and the two tangents with the two radii drawn from their extremities to the centre of the circle form a square. NOTE. We will now deduce some practical Problems from the foregoing Theorem. PROBLEM VIII. To draw a tangent to a circle from any given point. This Problem admits of two ca ses. COROLLARY I.--Hence, if from the extremities of any chord two tangents are drawn, they will form an isosceles triangle, whose equal angles CASE 1. When the point is in the circumference of the circle. (at the base) are measured by half the angle the chord subtends, and Let A be the given point from the angle at the vertex is measured which it is required to draw a tanby the supplement of the arc to which gent. Let A be the given point situated without the circle; BD, to which we are required to draw a tangent. From A draw the line ADC to the centre of the circle; then bisect the line AC (or divide it into two parts) in D (by Problem VII. Part II.;) then, with DA or DC as radius, describe the semicircle ABC, and where this cuts the circle BD in B, draw the line BA; then is BA a tangent to the circle BD, and drawn from the point A, as required. NOTE. We may here observe, that the truth of this Problem is manifest, for the angle BAC in the first case, and ABC in the second, is the angle in a semicircle, and, consequently, (by Theorem III. Part II.) it is a right angle, or the lines are square to one another; and (by Theorem V. Part II,) when that is the case, the line AB is a tangent to the circle. I shall here also take occasion to remark, that though AB is, strictly speaking a tangent to the point B, (see last fig.,) yet, if AB is produced to E, the whole line is mechanically understood to be the tangent line to the point B of the circle; but when the tangent is used as a line for the purpose of comparing the relative value of the sides of triangles, for the purposes of mensuration, &c. the line AB is always understood to terminate at the circumference in the point B, and is said to bẹ a tan MECHANICAL GEOMETRY. SIR,-In this age of inventions. when so many new things are presented for public consideration, per mit me to offer an old one in a new dress, which I think may be used to great advantage. There is nothing new in the ideas I have to suggest, so far as they relate to principles of science, and yet there may be in the applicationTwenty-four years since, when I first conceived the theory of this thing, I had in contemplation a trip across the Atlantic, at some distant day, and a series of experiments and demonstrations. But, circumstances admonish me that this hope should no longer be indulged. I have partially tested the plan, and if others can make it useful, and be the means of doing good, let them make the most of it. A small apparatus, at a trifling expense, may be so constructed as to produce a very bright light, which may be thrown upon the clouds in any direction, and made visible for many miles, in all directions. Line a semisphere with metallic or other reflectors, so as to receive the light from a number of lamps. and reflect it in any direction, the lights and reflectors turning on a common pivot. In a night of common darkness, a dozen good lamps will enable the operator to exhibit his light in the clouds, perfectly visible,.. at distances of from ten to twelve miles. I have done this in a machine that cost less than fifty dollars, so that the light has been seen near a hundred miles distant from the ope rator. Ships at sea, in distress, could throw signals, in this way, to such 187 distances as almost always to reach the eye of somebody, and gain relief. In very dark nights the effect is astonishing. Telegraphic signals may be so arranged, that ships crossing the Atlantic may speak others at immense distances, and report them on their arrival, always a day's sail in advance. The water may be examined, all around a vessel, with such an apparatus, at any hour of night, almost as minutely as by daylight. Letters of the alphabet, various figures of animals, &c. may be exhibited in the clouds, many miles distant, and changed in an instant, as well the figure as the position, over a space equal to the whole extent of the power of the light. ing from New York, goes fitted out Suppose a Liverpool packet, sailwith an apparatus of this kind, calculated for a focal distance of one to two or three miles Let her exhibit her light every night, going and returning, in the, clouds and the air oof vessels near her track that would ver her course, The great number report the strange lights they had seen, in all the ports where they arrive, would demonstrate the use that might be made of this contrivance, and that the theory is no ingenious fallacy. The power of the light may be made to extend sixty miles, or more; but the shorter its range, in the first experiment, the less will be the expense. Suppose she is approaching some coast, in distress; the signal be ing known, she gives her call for help, and her own exact position, in so wide a range, that relief may come in season. God grant that relief may come.- It may not be amiss to add, that, during the late war, when the inhabi tants of a certain maritime district New-York, March 23, 1825. 188 JUNCTION OF THE PACIFIC AND ATLANTIC OCEANS. AND ATLANTIC OCEANS. PROJECTED JUNCTION OF THE PACIFIC Venience of man, afford an abundant supply of water, and the course of the upper part of the river San Juan is capable of being diverted so as to constitute a never-failing feeder. water A project for opening a communication between the Atlantic and Pacific oceans, through the provinces of Darien, Biriquete, and Zi tara, is seriously agitated; and from the splendour and importance of the object, the enterprise and ability of the undertakers, and the liberal and enlightened policy pursued by the government of Colombia, we doubt not it will be very speedily accomplished The route is one of those named as exceedingly practicable by the Baron Humboldt, and as having been navigated more than fifty years since, with petiaugers, loaded with the produce of the Pacific shores, during the wet seasons of the year. It is through the Rio (river) Atrato and Rio San Juan; the former of which debouches into the bay of Candelaria, in the gulf of Darien and is navigable for vessels drawing seven feet water about four hundred miles, to the city of Zitara, and from thence through the Rio Quito to the Tombo S. Pablo. Though this navigation is interrupted by some falls or rapids, it is capable of being improved, for four feet draught of water, at no very great expense. The San Juan is also navigable for boats drawing five or six feet water as far up as the river Tamana or to within eighteen or twenty miles of the Tombo S. Pablo. The mouth of this river does not, however, afford a good harbour, though there is a road stead with very excellent anchorage. Proceeding southwardly, however, we soon come to the bay of Buenaventura, which is one of the best harbours in the Pacific cean, and which may be connected, through very favourable ground, by a cut of twelve miles in length, to the Rio Calima, which flows in almost a direct course, into the Rio San Juan. The summit level between the last named river and the Atrato does not exceed twenty-four feet, being alluvial and free from rocks, and the site is truly favourable for a canal. The lofty ranges of the neighbouring mountains, which, in this piace, seem to have been intercepted for the con These rivers, particularly the Atrato, are rapid, the current flowing in many places at the rate of three or four miles per hour; but the navigation, unlike that of the MississipPi, is not rendered dangerous by drift wood, planters and sawyers. The distance from ocean to ocean in the direction of the rivers, and the proposed canal is computed at about seven hundred and fifty miles, and it is expected that steam boats, constructed in nearly the form of our canal boats, and drawing from three to four feet water, will run it on an average in seven or eight days, and there is no Such is the inscarcity of fuel formation we have collected on this highly interesting subject; and, from the respectability of the sources from which it has been received, we feel satisfied of its correctness. The execution of this measure will unquestionably produce a revolution in a portion of the commerce of the world; the western shores of our con tinent will be virtually approximated to the eastern, and even to Europe; new marts of commerce will spring into being; steam boats will disturb and ruffle the surface of the Pacific ocean; and the tea, spices, silks, &c. of the Indies, the oil, fish, and furs of the North-west coast, and the precious metals of Peru, will, instead of following the circuitous and dangerous routes now traversed, find their way almost directly to our markets A new era will commence, and too nations now the citizens of often armed in hostile array against each other, will from frequent intercourse, become acquainted with the peculiarities of each other, and associate like brothers of the same community. Such at least is the prospective of this magnificent work, and such we trust in Heaven will be the reality FULTON. New-York, April, 1825; **A transcript of the contemplated route may be seen at J. V. Seaman's, Bookseller, Broadway. 1 BREWSTER'S WOOL SPINNING MACHINE. BREWSTER'S WOOL SPINNING MACHINE Extract from a memorial presented to the Gilbert Brewster, Esq, of the State of Connecticut, has invented a machine for spinning wool, which he has exhibited in this city, and is now to be seen in operation in the senate lobby, which as a labour saving maehine far surpasses any thing of the kind I have ever seen in this or any other country. I have seen it in operation, and have carefully and minutely calculated the difference of expense in spinning between this machine and the hand jennies now in use in our factories and from a practical knowledge of the business, I have no hesitation in saying that there is a saving of labour, equal at least to 60 per cent. This machine can be propelled by water, steam, or hand power; it spins from the rools, consequently supersedes the necessity of roping; in experienced boys or girls can attend it as well as the most experienced spinner, and I am fully convinced that the use of these machines in our woollen factories will afford a protection equal to an additional duty of 15 per cent. on foreign cloths. The object of your memorialist is to submit to the consideration of your honourable body the policy and propriety of adopting the use of those machines in your state prisons and county poor houses, whereby indigent women and children, and such of your convicts as are not sentenced to solitary confinement, can be profita bly employed; and I have no doubt that instead of your state prisons being a constant and heavy drain upon your treasury, they would become a very considerable source of revenue. With a view then to effect this desirable object, it would be indispensa bly necessary, in order to reap the extent of the benefits to be derived from this plan, to locate the new state prison which is contemplated to be built, as well as your county poor houses, in such place and places where sufficient water power can be applied to propel the machinery. 189 In adopting this plan, your honourable body will readily perceive, that you not only establish two great state woollen factories, but also one in each county within the state, where poor houses are required to be built, and thereby give profitable employment to your paupers and convicts This subject is also in a national point of view, one of great importance, as our national government is becoming sensible that the true policy of all wise governments is to promote and encourage the arts and sciences, as well as every species of home industry, as objects of primary importance, in all free and well regulated governments: for it must be admitted, that the wealth of a country depends altogether on the industry and enterprise of its citizens, and the acts of the government will never fail to secure the affections of the people. New York has immortalized herself by her great canals, which have given facility to her intercourse, and activity to her commerce. She has taken the lead in the great field of internal improvements; let her preserve her proud pre-eminence among her sister states She has put her hand to the plough, let her not look back; she has done much, and much yet remains to be done. Every step advanced accelerates the force, and multiplies the means, for human ingenuity has no assignable limits.The ocean of moral resources has never been fathomed, and is perhaps without shore or soundings. Our citizens are intelligent, inventive and enterprising, and there is none, perhaps, more so than Mr. Brewster; who. by his ingenuity and skill has contributed very materially to advance the useful arts, by which, society has been enabled to substitute for natural labour, the employment of that, which, being furnished by artificial contrivance, is less expensive The attention of the American people is anxiously directed to the policy of encouraging manufactures from domestic materials, as the most practicable mode of speedily advancing the substantial and permanent good of all classes, as all governments |