4G 



THE POPULAE, EDUCATOR. 



with a praiseworthy degree of patient labour ; but when all ia 

 done that he is able to do, his copy proves to be a failure in 

 some essential points. It is out of proportion ; the perspective 

 lines are not given correctly ; the curve-lines may not be zig- 

 zag, but they want the easy sweep of his exemplar ; they are 

 \ full of shoulders and joints ; and the perpendiculars are not 

 upright, nor are the horizontal lines at right angles to them. 

 When the first ardour of execution has abated, he perhaps dis- 

 covers these faults himself; and if he makes the common 

 mistake of supposing that the art of drawing is a gift, and that 

 the pencil is a magician's wand, manifesting its powers only in 

 the hand of some rightfuJ owner, he may then lose heart, and 

 think that his faculties are not adapted for the pursuit of this 

 noble art. If any of our readers have unfortunately stopped 

 at this point of their studies, let them recover their confidence, 

 and prosecute their favourite pursuit under our guidance. The 

 good method of practice, and the intelligible principles which 

 we propose to explain and set before them, will so lead their 

 hand and their eye, that ultimately they will accomplish all 

 their desire. 



Well-directed application does wonders in other arts, and why 

 not in drawing? What exercises does not a musician or a singer 

 go through, before he gets command of his voice or fingers ? 

 Who expects to arrive at that dazzling rapidity of motion visible 

 in the touch of the violin-player, certain and instantaneous 

 though it be, by any other method than that of hard and con- 

 stant practice ? Would not he who should begin to learn the 

 use of any instrument by attempting complete airs, and always 

 turning aside from the exercises which a master prescribes, be 

 sure to end where he began, and become no player ? Think 

 what an amount of labour is necessarily expended in fingering 

 by the young pianoforte-player. Greatly less labour than is 

 necessary in prosecuting many othor arts will make an able 

 draughtsman, fit him for the performance of many useful works, 

 and imbue him with those principles of drawing which are 

 applicable throughout the whole range of this art. 



It is frequently asserted that the art of drawing, like that of 

 writing poetry, is a natural gift; and that unless you possess 

 this, you never can excel. It may be true that, to rise to the 

 highest eminence in any science or art, requires a peculiar bent 

 of the mind ; but to acquire a useful practical knowledge of the 

 art of drawing, it is by no means necessary that every one should 

 be a genius. With regard to the sister arts poetry and painting 

 it may be truly said, in regard to their elements, at least, 

 that every man is endowed with some ability for their acquisition 

 and their application. Every one, for instance, is poetical when 

 he speaks on a subject with which he is well acquainted, or in 

 which he is deeply interested ; and, in like manner, every one is 

 an artist, who is ready to make a sketch or a drawing of any 

 object, which he wishes to explain to another, when he finds 

 that language fails to convey his ideas. The art of drawing, 

 therefore, may be attained to a sufficient extent for practical 

 purposes by every one who exerts the necessary attention and 

 assiduity. The artisan, the tradesman, or the connoisseur, may 

 by the use of a few well-directed strokes of the pencil, convey 

 on idea of his plans, operations, and views in relation to artistic 

 productions, of which the most laboured and elegant composition, 

 consisting of many hundred words, would fail to convey the 

 slightest impression to the mind of the hearer or the reader. 



LESSONS IN ARITHMETIC.- 



SUBTEACTION. 



-III. 



A. IF a leas number be taken away from a greater, or, aa it is 

 called, subtracted from it, the number left behind is called the 

 difference of the two numbers, or the remainder. 



The sign (called minus) placed between two numbers 

 indicates that the one before which it stands ia to be subtracted 

 from the other. 



2. When the individual figures composing the larger number 

 are respectively larger than the corresponding figures of the 

 smaller number, the process ia evident. We have only to take 

 the differences of the numbers of units, tens, hundreds, etc., 

 respectively, and the resulting number can be at once written 

 down. Thus, for instance, suppose it be required to find the 

 iifference between 9876 and 7653. 



Write down the numbers one under the other, the units 



under the units, the tens under the tens, the hundreds under 

 the hundreds, and so on, thus : 

 .9876 

 7653 



2223 

 3 units { in nu h ^ e r 88 } taken from 6 units { j^ } leave E 3 units. 



5 tens 



6 hundreds 



7 thousands 



7 tens 



8 hundreds 



9 thousands ,, 



2 tens. 



2 hundreds. 



2 thousands. 



Thus, the difference is 2 thousands, 2 hundreds, 2 tens, and 3 

 units, or, as it is written, according to the rules of our notation 

 2223. 



3. But suppose that the figures in the less number are not 

 respectively less than the corresponding figure in the other 

 number ; we must then proceed somewhat differently. 



The method we employ depends upon the following self- 

 evident proposition, or 



Axiom. If two numbers be increased by the same quantity, 

 their difference will not be altered. 



4. Suppose that it be required to subtract 4789 from 5231. 

 Place the numbers, one under the other, as before 



5231 

 4789 



442 



9 units in the less cannot be taken from 1 unit of the greater ; 

 add, however, 10 units to the 1 unit in the upper, and add 10 

 to the lower number by changing the 8 in the tens place into a 



9. The numbers are now 5 thousands, 2 hundreds, 3 tens, 

 and 11 units; and 4 thousands, 7 hundreds, 9 tens, and 9 units. 

 Now, 9 units from 11 units leave 2 units. 



Again, 9 tens cannot be taken from 3 tens, but if we increase 

 the 3 in the tens place of the upper number by ten, and the 

 7 in the hundreds place in the lower by one, we shall be adding 

 the same quantity (a hundred) to each number, since any figure 

 indicates a number ten times as great as the same figure in a 

 place immediately on its right. 



Then 9 tens from 13 tens leave 4 tens. 



Again, 8 hundreds cannot be taken from 2 hundreds, but if 

 we increase the 2 in the hundreds place of the upper number by 



10, and the 4 in the thousands place in the lower number by 1, 

 we shall be adding the same quantity (a thousand) to each 

 number, for the reason we have already mentioned above. 



Then, 8 hundreds taken from 12 hundreds leave 4 hundreds. 

 And 5 thousands from 5 thousands leave nothing. 



Hence the difference of the numbers is 4 hundreds, 4 tens, 

 and 2 units ; that is, 442. 



* 5. The process may also be clearly exhibited as follows : 



5231 = 5 x 1000 -*- 2 x 100 + 3 x 10 + 1 

 4789 = 4 x 1000 + 7 x 100 + 8 x 10 + 9 



The difference between these numbers is the same as the 

 difference between 



5 x 1000 + 12 x 100 + 13 x 10 + 11 

 and 5 x 1000 + 8 x 100 + 9 x 10 + 9 



For we have added the same quantity to the original numbers, 

 namely : 



10 x 100 + 10 x 10 + 10 i.e., 1110 to the upper, 

 and 1000 + 100 + 10 i.e., 1110 to the lower. 



The difference is clearly seen to be, therefore 

 4 x 100 + 4 x 10 + 2 



i.e., according to the principles of notation, 442. 



6. From the above analysis of the process of subtraction will 

 be perceived the truth of the following 



Rule for Subtraction. Write the less number under the 

 greater, so that units may stand under units, tens tinder tens, 

 etc. Beginning at the right hand, subtract each figure in the 

 lower number from the figure above it, and set down the re- 

 mainder directly under the figure subtracted. When a figure in 

 the lower number is larger than that above it, add 10 to the 

 upper figure ; then subtract aa before, and add 1 to the next 

 figure in the lower number. 



* Articles 5 and 7 may be omitted until after the lesson on Multi- 

 plication has been read. 



