Heights and Distances.

New Zealand Mail, Issue 942, 21 March 1890, Page 9

Heights and Distances.

Simfee Methods by Which They May be Easily Ascertained.

There are doubtless a large number of intelligent persous in every community, who, though not particularly interested or well versed iu the study of mathematics, have often felt the importance of knowing some convenient and; simple method for determining the height of a tree or the width of a stream, and to whom bucli knowledge would be very useful and desirable. It is related of Dr. Livingstone, the famous explorer, that when travelling in the wilds of Africa, he first oame in view of the magnificent Viotoria Falls,' he l'ouud himself without instruments ; but the most provoking trial to him was that he had f rgotten the simple mathematical rules of his schoolboy days, and in great sorrow of heart was obliged to turn away from this ■ beautiful river without being able to calculate or even ‘guess’ its dimensions. The erroneous estimates wbich are inatleof the height of trees, bnildings and other'objects, render desirable an easy and convenient method for measuring them ; and many persons with a ‘liberal education have doubtless found themselves in the nnpleasant predicament of Dr. Livingstone, and regretted -their irability to determine the height of some interesting object, or the distanoe across a river.

The height of a tree may be estimated sufficiently exact to? ordinary purposes by the following method. Being in the vicinity of a tree, the height of which yon may wish to know, and in. your hand you carry a walking-cane or a jointed fishing rod, and supposing the cane, or a length of the rod, is just three feet, set it in the ground vertically, anbif the sun shines, it will casta shadow; now, with a pocket-rule, you measure the length of .the shadow, and find it, say two feet. Here, then, we have a right angle of two feet and three feet. Now measure from the base of the tree.to the end of its shadow, and we will suppose it to be twenty feet. The problem, therefore, is simply this. If a cane thi-ee feet high casts a shadow of two feet, how high' must a tree be to cast a shadow of twenty feet ? Or, in other words, if two gives.’three, how much will twenty give ? By the simple ‘ rule of three ’we find the answer', to be thirty feet. Thus, by similar triangles, we have 3 :3 ::20 :X.\X = 30 feet—the tree’s height. There is another method, which has the advantage of being still more simple and convenient, by which the height of a tree may be easily determined by ite shadow. Any pe son ’may easily measure the exact height of a tree when the sun shines, or during bright moonlight, by making two lines on the-ground, three feet apart, and then placing in the ground, on the line nearest the sun, a stick that shall stand exactly three-feet but of the soil. When the end of the shad civ of the stick exactly touch the faithest line, then also the shadow of the tree will be exactly in length the same measurement as its height. Of course, in such a case, the sun will be at an exact angle of 45deg, or just midway between the zenith and the horizon.

But the reader may now ssk, ‘ Suppose the nun doesn’t shine, what then ?’ Why, then set up the cane as before, say eighteen feet from the base of the tree. Now placo your head on the ground with the cane between you and the tree, moving nearer to or farther from it, until you can just see the )op of the tree over the top of the cane ; then place a pebble or mark on the ground at the point where you obtain this view. The cane being three feet high, the distance from the pebble to it will be two feet, and from the pebble to the base of the tree, twenty feet, hence, by the same rule, we find the height of the tree to be thirty feet, as explained above. The following method, with a little practice will enable any person to measure the heights of trees or other objects with approximate acouracy when the sun is not Bhining, and the method here given represents the Bimples; and quickest way to measure heights, though the results are not absolutely corre.t.

First make a mark oa the tree or other object, say six feet from the ground, or place a pole six feet upright against it. Then walk away to such a distanoe that the breadth of the hand, held out at full arm’s length, will just cover the six feet. Mark with the eye a point on the tree at tbenpper end of the six feet, and move the hand upwards another breadth, and thus prooeed until the whole, height is measured! It may sometimes be convenient for an assistant to stand at the foot of the tree, and if with bis hat on he will be six feet high, he may serve as a measure to begin with instead of the rod. It is well to stand at some distance from the tree in making these measurements, or otherwise the upper measured portions will be larger than the lower on account of the ‘longer legs’ of the imaginary triangle. If the distance be too great for the breadth of the hand, one or two fingers only may be use!, op a short pocket rule. Or, if the pocket ruin bn used, its separate subdivision into inches may be made to indicate the portions measured, and the whole completed at one measurement.

The heights of perpendicular banks of lakes or other preoipices, or the descent of a water-fall, have been singularly misjudged for the want of some such means of measure, ment as those described above. If the water of a lake freezes in winter, the ice forms an excellent base-line for the measurement of any of its shores or banks, and the tops of trees which grow upon them. Some writer his pnt into rhyme an interesting problem in height-measurement, as given below, and the reader may determine the answer, which may be easily found by a simple rule in proportion. * ' If with a measure three feet long, The shadow of five is made. What is the steeple’s height in yards, That’s ninety feet in shade.’ , The following is an excellent method for determining the distance to any visible object, or the width of a river, by the aid of similar right-angled triangles : In the accompanying diagram, suppose a man at A desires / to know the distance to any object, as B, on the opposite side of the river. How should he proceed to determine this distance with the aid of a ten-foot pole ?

Form the right angle A, or B AC, and measure any distance, as A 0= 30 feet. Then from C measure any distance, as C D equals 9 feet; also D E=l2 feet perpendicular to C. A. Then by means of similar triangles we have C D : C A ::JD E: AB. That is, 9 : 30 :: 12 : A B ; hence A B, or the distance aoross the river =/40 feet. The mathematical principles involved in determining the ' distance o£. i >a» J star are essentially the same as..tHo'Se employed by the surveyor when he wishes to measure the width of a stream which he cannot cross ; only, with the astronomer the problem is rendered, much more complicated from the fact that he is unable to procure a ‘ base line ’ ot sufficient length to meet his requirementa; and even the diameter of the earth’s orbit is not long enough to use in measuring the distances of the stars, with very few excep. tions.

To learn the distance of a star, it is first necessary to determine what is known as a star’s ‘parallax,’ or its angle of direction when viewed from two opposite points in the earth’s orbit, and this is what renders the problem so extremely difficult ; for nearly every star in the heavens that, has been examined for the purpose of learning its distance, has failed to show y any parallax whatever, and in the few. instances . where a parallax has been recognised, the angle is found to be exceedingly small. No star in the heavens has a parallax equal to one Becond of arc, but all thus far determined are below even this small angle. The value of a second is so small that spider’s web, placed in the field of view of a telescope, completely hides the portion of the celestial sphere where the apparent movements of the stars are effected, a portion at most equal to not quite one second. It will thus be seen that the problem which astronomers have to solve in ascertaining the distances of the stars is one of the most stupendous difficulty. A parallax of one second means that the celestial object i 3 206,265 times farther away than we are from the sun, the Bun’s distance being one-half of the * base-line.’ If, then, a star’s parallax be less than one second, the star must be farther away than 206,265 times 93,000,000 miles, and this we find to be the case with every star in the heavens. In order that the length of any straight line seen shall be reduced so as to appear under the small angle of only one second, it is necessary that this line be at a distance of 206,255 times its length. Prof. Young, in bis excellent work on ‘ The Sun ’ says ‘ This number, 206,265, is the length of the radius of a circle expressed in seconds of its circumference. A ball One foot in actual diameter would have an apparent diameter of one second at a distance of 206,265 feet, or a little more than thirty-nine miles. If its apparent diameter were ten. seconds its distance would, of course, be only one-tenth as great.’—Arthur K. Bartlett* Battle Creek, Mich.