+----+----+----+----+----+----+----+----+----+----+----+ |2016|4212|1656|3852|1296|3492| 936|3132| 576|2772| 216| +----+----+----+----+----+----+----+----+----+----+----+ | 252|2052|4248|1692|3888|1332|3528| 972|3168| 612|2412| +----+----+----+----+----+----+----+----+----+----+----+ |2448| 288|2088|4284|1728|3924|1368|3564|1008|2808| 648| +----+----+----+----+----+----+----+----+----+----+----+ | 684|2484| 324|2124|4320|1764|3960|1404|3204|1044|2844| +----+----+----+----+----+----+----+----+----+----+----+ |2880| 720|2520| 360|2160|4356|1800|3600|1440|3240|1080| +----+----+----+----+----+----+----+----+----+----+----+ |1116|2916| 756|2556| 396|2196|3996|1836|3636|1476|3276| +----+----+----+----+----+----+----+----+----+----+----+ |3312|1152|2952| 792|2592| 36|2232|4032|1872|3672|1512| +----+----+----+----+----+----+----+----+----+----+----+ |1548|3348|1188|2988| 432|2628| 72|2268|4068|1908|3708| +----+----+----+----+----+----+----+----+----+----+----+ |3744|1584|3384| 828|3024| 468|2664| 108|2304|4104|1944| +----+----+----+----+----+----+----+----+----+----+----+ |1980|3780|1224|3420| 864|3060| 504|2700| 144|2340|4140| +----+----+----+----+----+----+----+----+----+----+----+ |4176|1620|3816|1260|3456| 900|3096| 540|2736| 180|2376| +----+----+----+----+----+----+----+----+----+----+----+ 35. If two numbers must be added together, it will not alter the sum if you take away a part of one, provided you put on as much to the other. It is plain that you will not alter the whole number of a collection of pebbles in two baskets by taking any number out of one, and putting them into the other. Thus, 15 + 7 is the same as 12 + 10, since 12 is 3 less than 15, and 10 is three more than 7. This was the principle upon which the whole of the process in (29) was conducted. 36. Let _a_ and _b_ stand for two numbers, as in (24). It is impossible to tell what their sum will be until the numbers themselves are known. In the mean while _a_ + _b_ stands for this sum. To say, in algebraical language, that the sum of _a_ and _b_ is not altered by adding _c_ to _a_, provided we take away _c_ from _b_, we have the following equation: (_a_ + _c_) + (_b_ - _c_) = _a_ + _b_; which may be written without brackets, thus, _a_ + _c_ + _b_ - _c_ = _a_ + _b_. For the meaning of these two equations will appear to be the same, on consideration. 37. If _a_ be taken twice, three times, &c., the results are represented in algebra by 2_a_, 3_a_, 4_a_, &c. The sum of any two of this series may be expressed in a shorter form than by writing the sign + between them; for though we do not know what number _a_ stands for, we know that, be it what it may, 2_a_ + 2_a_ = 4_a_, 3_a_ + 2_a_ = 5_a_, 4_a_ + 9_a_ = 13_a_; and generally, if _a_ taken _m_ times be added to _a_ taken _n_ times, the result is _a_ taken _m_ + _n_ times, or _ma_ + _na_ = (_m_ + _n_)_a_.