CUBE

, a regular or solid body, consisting of six equal sides or faces, which are squares.—A die is a small cube.

It is also called a hexaedron, because of its six sides, and is the 2d of the five Platonic or Regular bodies.

The cube is supposed to be generated by the motion of a square plane, along a line equal and perpendicular to one of its sides.

To describe a Rete, or Net, for forming a cube, or with which it may be covered.—Describe six squares as in the annexed figure, upon card paper, paste-board, or the like, of the size of the faces of the proposed cube; and cut it half through by the lines AB, CD, EF, AC, BD; then fold up the several squares till their edges meet, and so form the cube, or a covering over one, as in the figure annexed.

To determine the Surface and Solidity of a Cube.—Multiply one side by itself, which will give one square or face; then this multiplied by 6, the number of faces, will give the whole surface. Also multiply one side twice by itself, that is, cube it, and that will be the solid content.

Duplication of a Cube. See Duplication.

Cubes

, or Cubic Numbers, are formed by multiplying any numbers twice by themselves. So the cubes of 1, 2, 3, 4, 5, 6, &c, are 1, 8, 27, 64, 125, 216, &c.

The third differences of the eubes of the natural numbers are all equal to each other, being the constant number 6. For, let m³, n³, p³ be any three adjacent cubes in the natural series as above, that is, whose roots m, n, p have the common difference 1; then because ; so that the difference between the 1st and 2d, and between the 2d and 3d cubes, are and the dif. of these differences, is the 2d difference.

In like manner the next 2d dif. is : hence the dif. of these two 2d diffs. is , which is therefore the constant 3d difference of all the series of cubes. And hence that series of cubes will be formed by addition only, viz, adding always the 3d dif. 6 to find the column or series of 2d diffs, then these added always for the 1st diffs, and lastly these always added for the cubes themselves, as below:

3d Difs.	2d Difs.	1st Difs.	Cubes
6	6	1	0
6	12	7	1
6	18	19	8
6	24	37	27
6	30	61	64
6	36	91	125
6	42	127	216
6	48	169	343

Peletarius, among various speculations concerning square and cubic numbers, shews that the continual sums of the cubic numbers, whose roots are 1, 2, 3, &c, form the series of squares whose roots are 1, 3, 6, 10, 15, 21, &c.

It is also a pretty property, that any number, and the cube of it, being divided by 6, leave the same remainder; the series of remainders being 0, 1, 2, 3, 4, 5, continually repeated. Or that the differences between the numbers and their cubes, divided by 6, leave always o remaining; and the quotients, with their successive differences, form the several orders of figurate numbers. Thus,

Num.	Cubes.	Difs.	Quot.	1 Dif.	2 Dif.
1	1	0	0	0	0
2	8	6	1	1	1
3	27	24	4	3	2
4	64	60	10	6	3
5	125	120	20	10	4
6	216	210	35	15	5
7	343	336	56	21	6

The following is a Table of the first 1000 cubic numbers. |

TABLE OF CUBES.
Num	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes
1	1	60	216000	119	1685159	178	5639752	237	13312053	296	25934336
2	8	61	226981	120	1728000	179	5735339	238	13481272	297	26198073
3	27	62	238328	121	1771561	180	5832000	239	13651919	298	26463592
4	64	63	250047	122	1815848	181	5929741	240	13824000	299	26730899
5	125	64	262144	123	1860867	182	6028568	241	13997521	300	27000000
6	216	65	274625	124	1906624	183	6128487	242	14172488	301	27270901
7	343	66	287496	125	1953125	184	6229504	243	14348907	302	27543608
8	512	67	300763	126	200376	185	6331625	244	14526784	303	27818127
9	729	68	314432	127	2048383	186	6434856	245	14706125	304	28094464
10	1000	69	328509	128	2097152	187	6539203	246	14886936	305	28372625
11	1331	70	343000	129	2146689	188	6644672	247	15069223	306	28652616
12	1728	71	357911	130	2197000	189	6751269	248	15252992	307	28934443
13	2197	72	373248	131	2248091	190	6859000	249	15438249	308	29218112
14	2744	73	389017	132	2299968	191	6967871	250	15625000	309	29503629
15	3375	74	405224	133	2352637	192	7077888	251	15813251	310	29791000
16	4096	75	421875	134	2406104	193	7189057	252	16003008	311	30080231
17	4913	76	438976	135	2460375	194	7301384	253	16194277	312	30371328
18	5832	77	456533	136	2515456	195	7414875	254	16387064	313	30664297
19	6859	78	474552	137	2571353	196	7529536	255	16581375	314	30959144
20	8000	79	493039	138	2628072	197	7645373	256	16777216	315	31255875
21	9261	80	512000	139	2685619	198	7762392	257	16974593	316	31554496
22	10648	81	531441	140	2744000	199	7880599	258	17173512	317	31855013
23	12167	82	551368	141	2803221	200	8000000	259	17373979	318	32157432
24	13824	83	571787	142	2863288	201	8120601	260	17576000	319	32461759
25	15625	84	592704	143	2924207	202	8242408	261	17779581	320	32768000
26	17576	85	614125	144	2985984	203	8365427	262	17984728	321	33076161
27	19683	86	636056	145	3048625	204	8489664	263	18191447	322	33386248
28	21952	87	658503	146	3112136	205	8615125	264	18399744	323	33698267
29	24389	88	681472	147	3176523	206	8741816	265	18609625	324	34012224
30	27000	89	704969	148	3241792	207	8869743	266	18821096	325	34328125
31	29791	90	729000	149	3307949	208	8998912	267	19034163	326	34645976
32	32768	91	753571	150	3375000	209	9123329	268	19248832	327	34965783
33	35937	92	778688	151	3442951	210	9261000	269	19465109	328	35287552
34	39304	93	804357	152	3511808	211	9393931	270	19683000	329	35611289
35	42875	94	830584	153	3581577	212	9528128	271	19902511	330	35937000
36	46656	95	857375	154	3652264	213	9663597	272	20123648	331	36264691
37	50653	96	884736	155	3723875	214	9800344	273	20346417	332	36594368
38	54872	97	912673	156	3796416	215	9938375	274	20570824	333	36926037
39	59319	98	941192	157	3869893	216	10077696	275	20796875	334	37259704
40	64000	99	970299	158	3944312	217	10218313	276	21024576	335	37595375
41	68921	100	1000000	159	4019679	218	10360232	277	21253933	336	37933056
42	74088	101	1030301	160	4096000	219	10503459	278	21484952	337	38272753
43	79507	102	1061208	161	4173281	220	10648000	279	21717639	338	38614472
44	85184	103	1092727	162	4251528	221	10793861	280	21952000	339	38958219
45	91125	104	1124864	163	4330747	222	10941048	281	22188041	340	39304000
46	97336	105	1157625	164	4410944	223	11089567	282	22425768	341	39651821
47	103823	106	1191016	165	4492125	224	11239424	283	22665187	342	40001688
48	110592	107	1225043	166	4574296	225	11390625	284	22906304	343	40353607
49	117649	108	1259712	167	4657463	226	11543176	285	23149125	344	40707584
50	125000	109	1295029	168	4741632	227	11697083	286	23393656	345	41063625
51	132651	110	1331000	169	4826809	228	11852352	287	23639903	346	41421736
52	140608	111	1367631	170	4913000	229	12008989	288	23887872	347	41781923
53	148877	112	1404928	171	5000211	230	12167000	289	24137569	348	42144192
54	157464	113	1442897	172	5088448	231	12326391	290	24389000	349	42508549
55	166375	114	1481544	173	5177717	232	12487168	291	24642171	350	42875000
56	175616	115	1520875	174	5268024	233	12649337	292	24897088	351	43243551
57	185193	116	1560896	175	5359375	234	12812904	293	25153757	352	43614208
58	195112	117	1601613	176	5451776	235	12977875	294	25412184	353	43986977
59	205379	118	1643032	177	5545233	236	13144256	295	25672375	354	44361864

|

Num	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes	Num.	Cubes
355	44738875	417	72511713	479	109902239	541	158340421	603	219256227	665	294079625
356	45118016	418	73034632	480	110592000	542	159220088	604	220348864	666	295408296
357	45499293	419	73560059	481	111284641	543	160103007	605	221445125	667	296740963
358	45882712	420	74088000	482	111980168	544	160989184	606	222545016	668	298077632
359	46268279	421	74618461	483	112678587	545	161878625	607	223648543	669	299418309
560	46656000	422	75151448	484	113379904	546	162771336	608	224755712	670	300763000
361	47045881	423	75686967	485	114084125	547	163667323	609	225866529	671	302111711
362	47437928	424	76225024	486	114791256	548	164566592	610	226981000	672	303464448
363	47832147	425	76765625	487	115501303	549	165469149	611	228099131	673	304821217
364	48228544	426	77308776	488	116214272	550	166375000	612	229220928	674	306182024
365	48627125	427	77854483	489	116930169	551	167284151	613	230346397	675	307546875
366	49027896	428	78402752	490	117649000	552	168196608	614	231475544	676	308915776
367	49430863	429	78953589	491	118370771	553	169112377	615	232608375	677	310288733
368	49836032	430	79507000	492	119095488	554	170031464	616	233744896	678	311665752
369	50243409	431	80062991	493	119823157	555	170953875	617	234885113	679	313046839
370	50653000	432	80621568	494	120553784	556	171879616	618	236029032	680	314432000
371	51064811	433	81182737	495	121287375	557	172808693	619	237176659	681	315821241
372	51478848	434	81746504	496	122023936	558	173741112	620	238328000	682	317214568
373	51895117	435	82312875	497	122763473	559	174676879	621	239483061	683	318611987
374	52313624	436	82881856	498	123505992	560	175616000	622	240641848	684	320013504
375	52734375	437	83453453	499	124251499	561	176558481	623	241804367	685	321419125
376	53157376	438	84027672	500	125000000	562	177504328	624	242970624	686	322828856
377	53582633	439	84604519	501	125751501	563	178453547	625	244140625	687	324242703
378	54010152	440	85184000	502	126506008	564	179406144	626	245314376	688	325660672
379	54439939	441	85766121	503	127263527	565	180362125	627	246491883	689	327082769
380	54872000	442	86350888	504	128024064	566	181321496	628	247673152	690	328509000
381	55306341	443	86938307	505	128787625	567	182284263	629	248858189	691	329939371
382	55742968	444	87528384	506	129554216	568	183250432	630	250047000	692	331373888
383	56181887	445	88121125	507	130323843	569	184220009	631	251239591	693	332812557
384	56623104	446	88716536	508	131096512	570	185193000	632	252435968	694	334255384
385	57066625	447	89314623	509	131872229	571	186169411	633	253636137	695	335702375
386	57512456	448	89915392	510	132651000	572	187149248	634	254840104	696	337153536
387	57960603	449	90518849	511	133432831	573	188132517	635	256047875	697	338608873
388	58411072	450	91125000	512	134217728	574	189119224	636	257259456	698	340068392
389	58863869	451	91733851	513	135005697	575	190109375	637	258474853	699	341532099
390	59319000	452	92345408	514	135796744	576	191102976	638	259694072	700	343000000
391	59776471	453	92959677	515	136590875	577	192100033	639	260917119	701	344472101
392	60236288	454	93576664	516	137388096	578	193100552	640	262144000	702	345948008
393	60698457	455	94196375	517	138188413	579	194104539	641	263374721	703	347428927
394	61162984	456	94818816	518	138991832	580	195112000	642	264609288	704	348913664
395	61629875	457	95443993	519	139798359	581	196122941	643	265847707	705	350402625
396	62099136	458	96071912	520	140608000	582	197137368	644	267089984	706	351895816
397	62570773	459	96702579	521	141420761	583	198155287	645	268336125	707	353393243
398	63044792	460	97336000	522	142236648	584	199176704	646	269586136	708	354894912
399	63521199	461	97972181	523	143055667	585	200201625	647	270840023	709	356400829
400	64000000	462	98611128	524	143877824	586	201230056	648	272097792	710	357911000
401	64481201	463	99252847	525	144703125	587	202262003	649	273359449	711	359425431
402	64964808	464	99897344	526	145531576	588	203297472	650	274625000	712	360944128
403	65450827	465	100544625	527	146363183	589	204336469	651	275894451	713	362467097
404	65939264	466	101194696	528	147197952	590	205379000	652	277167808	714	363994344
405	66430125	467	101847563	529	148035889	591	206425071	653	278445077	715	365525875
406	66923416	468	102503232	530	148877000	592	207474688	654	279726264	716	367061696
407	67419143	469	103161709	531	149721291	593	208527857	655	281011375	717	368601813
408	67911312	470	103823000	532	150568768	594	209584584	656	282300416	718	370146232
409	68417929	471	104487111	533	151419437	595	210644875	657	283593393	719	371694959
410	68921000	472	105154048	534	152273304	596	211708736	658	284890312	720	373248000
411	69426531	473	105823817	535	153130375	597	212776173	659	286191179	721	374805361
412	69934528	474	106496424	536	153990656	598	213847192	660	287496000	722	376367048
413	70444997	475	107171875	537	154854153	599	214921799	661	288804781	723	377933067
414	70957944	476	107850176	538	155720872	600	216000000	662	290117528	724	379503424
415	71473375	477	108531333	539	156590819	601	217081801	663	291434247	725	381078125
416	71991296	478	109215352	540	157464000	602	218167208	664	292754944	726	382657176

|

Num	Cubes.	Num	Cubes.	Num	Cubes.	Num	Cubes.	Num	Cubes.
727	384240583	782	478211768	837	586376253	892	709732288	947	849278123
728	385828352	783	480048687	838	588480472	893	712121957	948	851971392
729	387420489	784	481890304	839	590589719	894	714516984	949	854670349
730	389017000	785	483736625	840	592704000	895	716917375	950	857375000
731	390617891	786	485587656	841	594823321	896	719323136	951	860085351
732	392223168	787	487443403	842	596947688	897	721734273	952	862801408
733	393832837	788	489303872	843	599077107	898	724150792	953	865523177
734	395446904	789	491169069	844	601211584	899	726572699	954	868250664
735	397065375	790	493039000	845	603351125	900	729000000	955	870983875
736	398688256	791	494913671	846	605495736	901	731432701	956	873722816
737	400315553	792	496793088	847	607645423	902	733870808	957	876467493
738	401947272	793	498677257	848	609800192	903	736314327	958	879217912
739	403583419	794	500566184	849	611960049	904	738763264	959	881974079
740	405224000	795	502459875	850	614125000	905	741217625	960	884736000
741	406869021	796	504358336	851	616295051	906	743677416	961	887503681
742	408518488	797	506261573	852	618470208	907	746142643	962	890277128
743	410172407	798	508169592	853	620650477	908	748613312	963	893056347
744	411830784	799	510082399	854	622835864	909	751089429	964	895841344
745	413493625	800	512000000	855	625026375	910	753571000	965	898632125
746	415160936	801	513922401	856	627222016	911	756058031	966	901428696
747	416832723	802	515849608	857	629422793	912	758550528	967	904231063
748	418508992	803	517781627	858	631628712	913	761048497	968	907039232
749	420189749	804	519718464	859	633839779	914	763551944	969	909853209
750	421875000	805	521660125	860	636056000	915	766060875	970	912673000
751	423564751	806	523606616	861	638277381	916	768575296	971	915498611
752	425259008	807	525557943	862	640503928	917	771095213	972	918330048
753	426957777	808	527514112	863	642735647	918	773620632	973	921167317
754	428661064	809	529475129	864	644972544	919	776151559	974	924010424
755	430368875	810	531441000	865	647214625	920	778688000	975	926859375
756	432081216	811	533411731	866	649461896	921	781229961	976	929714176
757	433798093	812	535387328	867	651714363	922	783777448	977	932574833
758	435519512	813	537366797	868	653972032	923	786330467	978	935441352
759	437245479	814	539353144	869	656234909	924	788889024	979	938313739
760	438976000	815	541343375	870	658503000	925	791453125	980	941192000
761	440711081	816	543338496	871	660776311	926	794022776	981	944076141
762	442450728	817	545338513	872	663054848	927	796597983	982	946966168
763	444194947	818	547343432	873	665338617	928	799178752	983	949862087
764	445943744	819	549353259	874	667627624	929	801765089	984	952763904
765	447697125	820	551368000	875	669921875	930	804357000	985	955671625
766	449455096	821	553387661	876	672221376	931	806954491	986	958585256
767	451217663	822	555412248	877	674526133	932	809557568	987	961504803
768	452984832	823	557441767	878	676836152	933	812166237	988	964430272
769	454756609	824	559476224	879	679151439	934	814780504	989	967361669
770	456533000	825	561515625	880	681472000	935	817400375	990	970299000
771	458314011	826	563559976	881	683797841	936	820025856	991	973242271
772	460099648	827	565609283	882	686128968	937	822656953	992	976191488
773	461889917	828	567663552	883	688465387	938	825293672	993	979146657
774	463684824	829	569722789	884	690807104	939	827936019	994	982107784
775	465484375	830	571787000	885	693154125	940	830584000	995	985074875
776	467288576	831	573856191	886	695506456	941	833237621	996	988047936
777	469097433	832	575930368	887	697864103	942	835896888	997	991026973
778	470910952	833	578009537	888	700227072	943	838561807	998	994011992
779	472729139	834	580093704	889	702595369	944	841232384	999	997002999
780	474552000	835	582182875	890	704969000	945	843908625	1000	100000000
781	476379541	836	584277056	891	707347971	946	846590536

|

The Cube of a Binomial, is equal to the cubes of the two parts or members, together with triple of the two parallelopipedons under each part and the square of the other; viz, . And hence the common method of extracting the cube root.

Cubic Equations, are those in which the unknown quantities rise to three dimensions; as x³=a, or , or , &c.

All cubic equations may be reduced to this form, ; viz, by taking away the 2d term.

All cubic equations have three roots; which are either all real, or else one only is real, and the other two imaginary; for all roots become imaginary by pairs.

But the nature of the roots as to real and imaginary, is known partly from the sign of the co-efficient p, and partly from the relation between p and q: for the equation has always two imaginary roots when p is positive; it has also two imaginary roots when p is negative, provided ―(1/3)p)³ is less ―(1/2)q)², or 4p³ less than 27q²; otherwise the roots are all real, namely, whenever p is negative, and 4p³ either equal to, or greater than 27q².

Every cubic equation of the above form, viz, wanting the 2d term, has both positive and negative roots, and the greatest root is always equal to the sum of the two less roots; viz, either one positive root equal to the sum of the two negative ones, or else one negative root equal to the sum of two smaller and positive ones. And the sign of the greatest, or single root, is positive or negative, according as q is positive or negative when it stands on the right-hand side of the equation, thus ; and the two smaller roots have always the contrary sign to q.

So that, in general, the sign of p determines the nature of the roots, as to real and imaginary; and the sign of q determines the affection of the roots, as to positive and negative. See my Tract on Cubic Equations in the Philos. Trans. for 1780.

To find the Values of the Roots of Cubic Equations. Having reduced the equation to this form , its root may be found in various ways; the first of these, is that which is called Cardan's Rule, by whom it was first published, but invented by Ferreus and Tartalea. See Algebra. The rule is this: Put a=(1/3)p, and b=(1/2)q; then is Cardan's root ; or if there be put , and ; then , the 1st or Cardan's root, also is the 2d root, and is the 3d root.

Now the first of these, or Cardan's root, is always a real root, though it is not always the greatest root, as it has been commonly mistaken for. And yet this rule always exhibits the root in the form of an imaginary quantity when the equation has no imaginary roots at all; but in the form of a real quantity when the equation has two imaginary roots. See the reason of this explained in my Tract above cited, pa. 407. As to the other two roots, viz, though, in their general form, they have an imaginary appearance; yet, by substituting certain particular numbers, they come out in a real form in all such cases as they ought to be so.

But, after the first root is found, by Cardan's rule, the other two roots may be found, or exhibited, in several other different ways; some of which are as follow:

Let r denote the 1st, or Cardan's root, and v and w the other two roots: then is , and vwr=q; and the resolution of these two equations will give the other two roots v and w.

Or resolve the quadratic equation , and its two roots will be those sought. Or the same two roots will be either.

Ex. 1. If the equation be : here ; hence : therefore , the 1st root; and , the other two roots.

Ex. 2. If : here a=-2, and b=2; therefore : hence then , the first root; and 1±√3 the other two roots.

Ex. 3. If : here a=6, and b=3; then , and : therefore , the 1st root, and are the two other roots.

2. Another method for the roots of the equation , is by means of infinite series, as shewn at pa. 415 and seq. of my Tract above cited; whence it appears that the roots are exhibited in various forms of series as follow: viz, &c for the 1st root, and &c for the two other roots: where , and . |

And various other series for the same purpose may also be seen in my Tract, so often before cited.

3. A third method for the roots of cubic equations, is by angular sections, and the table of sines. It was first hinted by Bombelli, in his Algebra, that angles are trisected by the resolution of cubic equations. Afterwards, Vieta gave the resolution of cubics, and the higher equations, by angular sections. Next, Albert Girard, in his Invention Nouvelle en l'Algebre, shews how to resolve the irreducible case in cubics by a table of sines: and he also constructs the same, or finds the roots, by the intersection of the hyperbola and circle. Halley and De Moivre also gave rules and examples of the same sort of resolutions by a table of sines. And, lastly, Mr. Anthony Thacker invented, and Mr. William Brown computed, a large set of similar tables, for resolving affected quadratic and cubic equations, with their application to the resolution of biquadratic ones.

4. Lastly, the several methods of approach, or approximation, for the roots of all affected equations, which have been used in various ways by Stevin, Vieta, Newton, Halley, Raphson, and others.

To these may be added the method of Trial-and-error, or of Double Position, one of the easiest and best of any. Of this method, let there be taken the last example, viz, , in which it is evident that x is very nearly equal to 1/3, but a little less; take it therefore x=.33; then , but should be 6, and therefore the error is .024063 in defect.

Again suppose x=.34; then , which is .159304 in excess. Therefore , the root as before very nearly.

For the construction of cubic equations, see CONSTRUCTION.

Cubic Foot, of any thing, is so much of it as is contained in a cube whose side is one foot.

Cubic Hyperbola, is a figure expressed by the equation xy²=a, having two asymptotes, and consisting of two hyperbolas, lying in the adjoining angles of the asymptotes, and not in the opposite angles, like the Apollonian hyperbola; being otherwise called by Newton, in his Enumeratio Linearum Tertii Ordinis, an hyperbolismus of a parabola; and is the 65th species of those lines according to him.

Cubic Numbers. See Cubes. Fig 2

Cubic Parabola, a curve, as BCD, of the 2d order, having two infinite legs CB, CD, tending contrary ways. And if the absciss, AP or x, touch the curve in C, the relation between the absciss and ordinate, viz, AP=x, and PM=y, is expressed by the equation ; or when A coincides with C, by the equation y=ax³, which is the simplest form of the equation of this curve.

If the right line AP (fig. 2) cut the cubical parabola in three points A, B, C; and from any point P there be drawn the right line or ordinate PM, cutting the curve in one point M only: then will PM be always as the solid APXBPXCP; which is an essential property of this curve.

And hence it is easy to construct a cubic equation, as , by the intersection of this curve and a right line. See the Construction of a cubic equation by means of the cubic parabola and a right line, by Dr. Wallis, in his Algebra: As also the Construction of equations of 6 dimensions, by means of the cubic parabola and a circle, by Dr. Halley, in a lecture formerly read at Oxford.

The curve of this parabola cannot be rectisied, not even by means of the conic sections. But a circle may be found equal to the Curve Surface, generated by the rotation of the curve AM about the tangent AP to the principal vertex A. Let MN be an ordinate, and MT a tangent at the point M; and let PM be parallel to AN. Divide MN in the point O, so that MO be to ON as TM to MN. Then a mean proportional between TM+ON and 1/3 of AN will be the radius of a circle, whose area is equal to the superficies described by that rotation, viz, of AM about AP.

The Area of a Cubic Parabola is 3/4 of its circumscribing parallelogram.

Cubic Root, of any number, or quantity, is such a quantity as being cubed, or twice multiplied by itself, shall produce that which was given. So, the cubic root of 8 is 2, because 2³ or 2X2X2 is equal to 8.

The common method of extracting the cube root, founded on the property given above, viz, , is found in every book of common arithmetic, and is as old at least as Lucas de Burgo, where it is first met with in print. Other methods for the cube root may be seen under the article EXTRACTION of Roots, particularly this one, viz, the cube root of n, or , very nearly, or the cube root of n nearly; where n | is any number given whose cube root is sought, and a³ is the nearest complete cube to n, whether greater or less.

For example, suppose it were proposed to double the cube, or, which comes to the same thing, to extract the cube root of the number 2. Here the nearest cube is 1, whose cube root is 1 also, that is, a³=1, and a=1, also n=2; therefore nearly.

But, for a nearer value, assume now ; then is ; hence , or the cube root of 2, which is true in the last place of decimals.

And this is the simplest and easiest method for the cube root of any number. See its investigation in my Tracts, vol. 1, pa. 49.

Every number or quantity has three cubic roots, one that is real, and two imaginary: So, the cube root of 1 is either 1, or ; and if r be the real root of any cube r³, the two imaginary cubic roots of it will be for any one of these being cubed, gives the same cube r³.

Cubing of a Solid. See Cubature.

Cubo-cube

, the 6th power.

Cubo-cubo-cube

, the 9th power.