From the foregoing discussion, it is made clear that as far as the total volume of the universe is concerned, both the traditions- Shvetambara as well as Digambara-have accepted that it is 343 cubic *rajjus*. Now, the difference in their opinions is regarding the exact shape of the universe. Both the traditions are having the same view with regards the following facts:

- The total height (one dimension) of the universe is 14
*rajjus*; each of the lower and the upper universes has the height of 7*rajjus*. - The base of the universe (i.e., the lower end of the lower universe) is 7
*rajjus*in both dimensions-length as well as breadth. - The length of the universe at the upper end of the lower universe is one
*rajju*. - The length of the universe in the middle of the upper universe is five
*rajjus*. - The length of the universe at the top of the universe is again one
*rajju*.

The main difference is regarding the breadth (i.e., the third dimension). Whereas the Digambara tradition asserts that this dimension is 7 *rajjus *throughout the universe, according to the Shvetambara tradition, this dimension is different at different heights. Thus, in the opinion of Digambaras, the shape of the universe is *āyata-caturasra ^{[1]}* (as given in fig. 4) whereas the Shvetambaras do not agree with this, as according to them, the breadth is not 7

*rajjus*throughout the universe. According to them, at any particular height, the length and breadth should be equal.

Before, we make a critical examination; let us make a survey of the works where we get the mathematical description.

In the Digambara tradition, we get such description in several works, some of which that are worth mentioning are: *Dhavalā* *Ṭ**ikā*, a commentary on Ṣ*a**ṭkha*ṇḍ*āgama *by Acharya Virasena, *Tiloyapa**ṇṇatti *by Yativrishabhacharya, *Jambudīva paṇṇatti Saṃgaho *by Padmanandi, *Harivaṃśapurāṇa *by Acharya Jinasena of *Punnāṭa Saṃgha*, Swāmī *Kārtikeyānuprekṣā *etc.

Now, there are two opinions amongst the scholars as to who was the propounder of the belief that one dimension (north-south breadth) of the universe is 7 *rajjus *throughout. According to one opinion, it was Acharya Virasena, the great mathematician of 9th Century A.D., who was the first to propound this view. Before him, in the Digambara tradition there was no such view at all. On the other hand, the second group of scholars does not accept this view and hold that the belief that the universe is 7 *rajjus *in one dimension is an ancient one.

Pt. Phoolachandji Siddhantashastri, who represents the first group, maintains.^{[2]} “It was the author of the *Dhavalā*, Virasenacharya, who first of all established the view that the universe was 7 *rajjus *breadth in north-south direction throughout.” Before him, we find this view nowhere, as it becomes clear from the earlier works like *Rājāvārtika *etc.. Even in *Tiloyapa**ṇṇatti*,^{[3]} this view, originally established by Virasenacharya, was adopted (later on). Therefore, the present text of *Tiloyapa**ṇṇatti* seems to be later than the time of Virasenacharya. The thing which is to be paid attention in this connection is that the original text of *Tiloyapa**ṇṇatti *was available to Virasenacharya, and still he has explicitly mentioned that he has proved the shape of the universe to be “*āyata-caturasrākrāra*” (as given in fig. 4) only on the basis of two ancient verses only through his own reasoning power (and not on the base of *Tiloyapa**ṇṇatti*).^{[4]} Had this view been clear before him in the original text of *Tiloyapaṇṇatti *as it is found in its present edition, then he need not have dwelt upon this topic as elaborately as he has done in *Dhavalā*-in *Jīva**ṭṭhaṇa Kṣetrānugama *from pp. 10 to 22, and also he would not have ventured to claim at p. 15 of *Sparaśānugama *that only by dint of his own logical genius he had established the testing procedure for the extent of *rajjuccheda *(division of *rajju*) in the same way as he had made innovative establishment concerning the duration of *antaramuhūrtta *measuring innumerable *āvalīes *(a type of time-unit) and the *yatacaturasrakāra loka *(the universe having the shape as mentioned in fig. 4). As far as the topic of *rajjuccheda *is concerned, he has already explicitly accepted that he had got the necessary substantiation in the *Tiloyapaṇṇatti *for his innovation; then there is no reason to believe that had he (in the same way) found any indication or base of the view of *āyatacatursrākāra loka *in the *Tiloyapaṇṇatti*, he would have refrained to give its reference too. For, he, in fact, needed such substantiation to support his innovative idea of *āyuatacatursrākāra*, but as he could not find any such supporting evidence, he had to satisfy only by saying that his new view, viz., ‘the universe in 7 *rajjus *in breadth north-south wise throughout’, was not at all against any accepted *karaṇānuyoga *theory, for in scripture we did neither get any statement against nor in favour of it.^{[5]} Thus, it becomes quite clear that there is total absence of this belief in the entire literature known to the author of *Dhavalā*. Even today, we do not get a single allusion to the above belief in the definitely known earlier literature; at least it goes beyond doubt that there is no such reference supporting the innovative view in the version of *Tiloyapaṇṇatti *then available to Virasenacharya.” (Here ends the quotation of Pt. Phoolachandji Siddhantashastri).

The eminent scholar Prof. Hiralal Jain, concluding his discussion about the time of composition of *Tiloyapaṇṇatti*, writes; ^{[6]} “The implication of our deliberation is that the *Tiloyapaṇṇatti *can be conjectured to be a work which must be later than *Lokavibhāga *by *Sarvanandī *and earlier than *Dhavalā *of Virasenacharya; it means its time of composition falls between *Śaka *380 and 738 (or *Vikrama Saṃvat *495 and 873). After modification, made from time to time, its new version emerged any time after the composition of *Dhavalā*.”

Now, on behalf of the opposite opinion; Pt. Jugalkishore Mukhtar has presented several evidence in favour of the belief that the view that “the shape of universe has the breadth of 7 *rajjus *throughout in north-south direction”, is ancient one in his article. The arguments put forward by Pt. Mukhtar have been refuted by the counter group. Prof. Hiralal Jain writes in this context:^{[7]}-

“The opposite view has been forwarded by Pt. Jugalkishoreji by giving three evidence to prove that the belief of the universe having the same width of 7 *rajjus *in north-south direction throughout is earlier than Virasenacharya. These three allusions are from

- Jinasena’s
*Harivaṃśa Purāṇa*^{[8]} *Svāmi Kārtikeyānuprek**ṣā**Jambūdvīpapraj**–apti.*”

Further, in refutation of the above three evidence, Prof. Hiralal Jain observes:^{[9]} “It is true that in *Harivaṃśapurāṇa*, the *loka *(universe) is called “*caturasraka*”, but there is no mention at all of the fact that it is south-north wise 7 *rajjus *in breadth throughout. The word *caturasraka *may also connote *Samacaturasraka*. If by calling the *loka caturasraka*, it is understood as *āyatacaturasraka*, then it would imply definitely that in *Harivaṃśa Purāṇa*, it was Virasena who has been revered as guru, it is he who has been given the appellation *kavi cakravartī *(poet-sovereign) and it is he whose serene fame has been referred to.^{[10]} Not only this, but also there the mention of Jinasena as the disciple of Virasena and his work *Parśvābhyudaya *is made.^{[11]} In such circumstance, how is it said that the reference in *Harivaṃśa Purāṇa *is earlier than Virasena and the author of that *Purāṇa *was not familiar with the work of Virasena. On the contrary the above reference proves that the author (Jinasenacharya) of *Harivaṃśa Purāṇa *was well familiar and influenced with the Virasena’s work.

“Accepted that in *Swami Kārtikeyānuprekṣā*,^{[12]} there is explicit view about the shape of *loka *being 7 *rajjus *in width in north south direction, but what about the time of composition of this work? Pt. Jugalkishoreji has only mentioned that it is a very ancient work and was composed several centuries before Virasenacharya, but Pt. Jugalkishoreji has given no evidence to prove its antiquity.^{[13]} In such case, the allusion in this work cannot be regarded as a proof of its view about universe being earlier that Virasena.

“In the *Jambūdvīpaprajapti*,^{[14]} this belief is explicitly mentioned. But, here again, the determination of the time of composition of this work is quite imaginary, without a definite evidence. Pt. Mukhtarji has himself observed, “If this conjecture is correct, the time of *Jambūdvīpaprajapti *would be about *Śaka *670, i.e., Vikrama Saṃvat 805. But, until this conjecture is definitely given the form of ascertainment, we cannot accept the antiquity of *Jambūdvīpaprajapti *with respect to *Dhavalā *on the basis of this conjecture. According to the author (Padmanandi) himself, the *Jambūdvīpaprajapti *was composed at Bārānagara of Pāriyātra Deśa in the time of King Shaktikumara.^{[15]} An edict of one Shaktikumara, a king of Guhilavaṃśa, dated Vaiśākha Sudi 1, Vikrama Saṃvat 1034, has been obtained at Āhaḍa (near Udaipur). Also two edicts of the same time are found at Jain temples, but the portion mentioning date has been got wiped. Padmanandi (the author of *Jambūdvīpaprajapti *had composed his work in the time of this king only.^{[16]}

Therefore, this work may be of eleventh century of Vikrama Era.”

As mentioned in the footnotes 2 and 3 (p. 36), the time of Kārtikeya (or Kumāra Svāmī) is definitely earlier than that of the author of *Dhavalā*. If this is accepted, then the argument forwarded in the above quotation by Dr. Hiralal Jain would not hold good. But, again here also it may be argued that as *Tiloyapaṇṇatti *is earlier than *Dhavalā*, still the allusion to the belief of 7 *rajjus *width is a later addition, in the same way although the work “*Svāmī Kārtikeyānuprekṣā*” is earlier than *Dhavalā*, the above belief is a later addition there too. This is supported by the fact that Umā Svāmī, the author of the *Tattvārtha Sūtra*, is stated to be almost contemporary or slightly later than Kumāra Svāmī^{[17]} and the above view (of 7 *rajjus *width) is not mentioned in *Tattvārtha Sūtra*. It means that this view did not at all exist in the time of Umā Svāmī. It, thus, appears that it was a later addition in the work of Kumāra Svāmī.

On the basis of the foregoing discussion, we may conclude that the belief that “the universe is 7 *rajjus *in breadth throughout in the north-south direction” seems to be the original innovative establishment of Ācārya Virasena and not an ancient belief. This is well corroborated by number of proofs and most of the scholars have accepted it. We may then conclude that before Virasena, the author of *Dhavalā*, there was no clear mathematical concept in the Digambara tradition. At that time, the concept of *mṛdañgākāra loka *was prevalent amongst the Shvetambara Acharyas, which is clearly mentioned in the Dhavalā. But the Shvetambara concept had a defect that the volume of *mṛdañgākāra loka *was much less than 343 cubic *rajjus*. Virasenacharya not only successfully calculated the volume of the *mṛdañgākāra loka*, which came out to be^{[18]} 164 328/1356 cubic *rajjus *but he, on the base of two ancient verses, also showed that the volume of the lower universe was 196 cubic *rajjus *and that of the upper universe was 147 cubic *rajjus*, thus making total volume 343 cubic *rajjus*. By dint of his mathematical genius, he found out that this was possible only if one dimension of the universe was assumed to be 7 *rajjus *throughout. On the basis of this assumption, he imagined the geometrical shape in such a way that the total volume came out to be exactly 343 cubic *rajjus *and the volumes of upper and lower universes came out to be 196 cubic *rajjus *and 147 cubic *rajjus *respectively. The specialty of this geometrical shape was that the fundamental beliefs, viz., that the *loka *was 14 *rajjus *in height, that it was 7 and 1 *rajju *in length respectively at the lower end and in the middle and that it was 5 and 1 *rajju *in length respectively in the middle and the upper ends of the upper universe, were kept intact. Thus, this shape of the universe conceived by Virasena was unanimously accepted in the Digambara tradition. (And later on, as conjectured by some scholars, it was corporated even in ancient works like *Tiloyapaṇṇatti*).

In the Shvetambara tradition, besides the above fundamental beliefs, there were two more:

- The length and breadth at a particular height of the
*loka*should be the same. - There should be a gradual increase or decrease in length and breadth as we go up or down. This means:

The middle *pratara *(plane) is 1 *rajju *in length and breadth; this is called the *k*ṣ*ullaka pratara *(the plane of the least dimensions). When we go downwards from this place, there will be a gradual increase in the dimensions of the universe till we reach the lowest end, where both length and breadth become 7 *rajjus*. When we go upwards from the *k**ṣullaka pratara*, there is a gradual increase in the dimensions of the universe up to the middle of the upper universe (near the Brahmaloka, which is the fifth heaven) where both dimensions become 5 *rajjus*, and from there onwards again a gradual decrease ensues up to the upper end of the universe, where both dimensions become only 1 *rajju*. This has been described thus: ^{[19]} “There is an increase of “innumerableth part of one *aṅgula*” in the lateral dimensions (length and breadth) of the universe when we go “innumerableth part of one *aṅgula” * in the upward direction from the *kṣullaka pratara*; the increase in lateral dimension is called *tiryak-vṛddhi *and the distance in the vertical direction is called *ūrdhvagata *distance.^{[20]} The *tiryak-vṛddhi *at any point is less than the *ūrdhvagata *distance. (Thus, if we denote the *tiryak-vṛddhi *of innumerableth part of *aṅgula *by a1 and the *ūrdhvagata *distance of innumerableth part of *aṅgula *by a2, then a1*rajjus *each; again, when we move upwards, the dimensions go on decreasing till we reach the upper end of the upper universe, where they (length and breadth) become only one *rajju*. In the downward motion from the second *kṣullaka pratara*, when the vertical distance travelled is 1/innumerableth *aṅgula*, there is corresponding increase in the length and breadth (lateral dimensions), till the lower end of the universe is reached where they are 7 *rajjus *each.”

If these two additional fundamental beliefs are also accepted, then the assumption that “the universe is 7 *rajjus *in breadth throughout in the north-south directions” becomes untenable, and hence, unacceptable to Shvetambaras. As far as the total volume is concerned, the Shvetambara Acharyas also are of the view that it should be 343 cubic *rajjus*. But they do not want to give up the fundamental beliefs. Thus, by keeping intact all the fundamental beliefs, Shvetambara Acharyas have made efforts to find out the shape of the universe with volume of 343 cubic *rajjus*.

In the ‘*Vargita Lokamāna *’(described on supra, p. 142), the Acharyas have made an effort to give a mathematical solution. The way in which they have divided the whole universe into 56 parallelopipeds seems to be a primitive form of modern method of integral calculus. The dimensional measures assumed in the *Vargita Lokamāna *are not the exact/actual dimensions of the universe, as they are not in conformity with a fundamental belief of the Shvetambaras, which requires that there should be only innumerableth part of an *aṅgula’s *increase/decrease in vertical as well as lateral dimensions, but in the *Vargita Lokamāna *the *khaṇ**ḍuka *is taken to be the unit of increase/decrease, and because *khaṇ**ḍuka *is ¼ *rajju*, the increase/decrease is not as required. (Actually *khaṇḍuka *is much higher than the ‘innumerableth part of the one *aṅgul*a ‘, which would require a continuous increase/ decrease whereas in *Vargita Lokamāna*, the same dimensions prevails for ¼ *rajju*). Therefore, it becomes clear that the shape assumed in *Vargita Lokamāna *is only conjectural and an approximation to actual shape; it is, in fact, an effort to find out the volume of a figure which is of a complex (curved) nature, only for making the calculation easier.

In Shvetambara works,^{[21]} we also find the belief which explicitly accepts that the total volume of the *loka *is 343 cubic *rajjus*, in which the volume of the lower universe is 196 cubic *rajjus *and that of the upper universe is 147 cubic *rajjus*. But, we do not get any mathematical method there to prove this. Only through approximation, it is tried to prove that the total volume is 343 cubic *rajjus*; this we have already discussed and found that in terms of exact mathematical computation, it is not precise.

We may conclude our critical discussion of both Digambara and Shvetambara traditional account of the shape of the universe by saying that although Virasena’s innovative contribution is mathematically perfect, it is not acceptable to Shvetambaras; the efforts of the Shvetambaras, on the other hand, do not render a mathematically acceptable shape.

- Harivaṃśa Purāṇa, part-1, śloka 39.

Desammi pāriyatte jiṇanavaṇavibhūsie ramme.. - Ibid., 13.166

*gharmāghanodadhighanatanuvātān vihāyasaḥ.*

asaṃkhyabhāgaṃ cātītya, madhya lokasya kīrtitam..

asmādūrdhvamadhaścaiva, saṃpūrṇaḥ sapta rajjavaḥ.

atha trayāṇāṃ lokānāṃ, pratyekaṃ madhyamucyate..

gharmāyāṃ sarvataḥ kṣullamatrāsti prataradvayam.

maṇḍakākāramaikaikakhapradeśātmakaṃ ca tat..

rycake’tra pradeśānāṃ, yaccatuṣkadvayaṃ sthitam.

tatsamaśreṇikaṃ tacca, vijñeyaṃ prataradvayam..

lokavṛddhirūrdhvamukhī, tayoruparisaṃsthitāt.

adhaḥsthitātpunastasmāllokavṛddhiradhomukhī..

asaṃkhyabhāgaṃ cātītya, madhya lokasya kīrtitam..

asmādūrdhvamadhaścaiva, saṃpūrṇaḥ sapta rajjavaḥ.

atha trayāṇāṃ lokānāṃ, pratyekaṃ madhyamucyate..

gharmāyāṃ sarvataḥ kṣullamatrāsti prataradvayam.

maṇḍakākāramaikaikakhapradeśātmakaṃ ca tat..

rycake’tra pradeśānāṃ, yaccatuṣkadvayaṃ sthitam.

tatsamaśreṇikaṃ tacca, vijñeyaṃ prataradvayam..

lokavṛddhirūrdhvamukhī, tayoruparisaṃsthitāt.

adhaḥsthitātpunastasmāllokavṛddhiradhomukhī..

- Lokaprakāśa, cantos XII, verses 49-53.

*Pūrṇaikarajjūpṛthulāt, kṣullakapratarāditaḥ.*

Ūrdhvaṃ gate’ṅgulāsaṃkhyabhāge tiryagvivarddhate..

Aṅgulasyāsaṃkhyabhāgaḥ, pāramatreti bhāvyatām.

Ūrdhvagādaṅgulasyāṃśādaṃśastiryaggato laghuḥ..

Evamadho’pi.

Evaṃ cordhvalokamadhyaṃ, pṛthuluaṃ pañca rajjavaḥ.

Hīyate’tastathaivordhvaṃ, rajjurekā’vaśiṣyate..

Kiñca -

Rajjumānād dvitīyasmāt, kṣullakapratarāccitiḥ.

Adhomukhī ca tiryak cāṅgulāsaṃkhyāṃśamātrikā..

Evaṃ cādholokamūle, pṛthutvaṃ sapta rajjavaḥ.

Ūrdhvaṃ gate’ṅgulāsaṃkhyabhāge tiryagvivarddhate..

Aṅgulasyāsaṃkhyabhāgaḥ, pāramatreti bhāvyatām.

Ūrdhvagādaṅgulasyāṃśādaṃśastiryaggato laghuḥ..

Evamadho’pi.

Evaṃ cordhvalokamadhyaṃ, pṛthuluaṃ pañca rajjavaḥ.

Hīyate’tastathaivordhvaṃ, rajjurekā’vaśiṣyate..

Kiñca -

Rajjumānād dvitīyasmāt, kṣullakapratarāccitiḥ.

Adhomukhī ca tiryak cāṅgulāsaṃkhyāṃśamātrikā..

Evaṃ cādholokamūle, pṛthutvaṃ sapta rajjavaḥ.

- Ibid., cantos XII, verses 99-103.

Ūrdhva Saṃvartaka:

(Ghana) | (Cubic) | khaṇḍukas | – | 9408 |

(Sūcī) | (Linear) | (rajjus) | – | 2352 |

(Pratara) | (Square) | (rajjus) | – | 588 |

(Ghana) | (Cubic) | (rajjus) | – | 147 |

Adho Saṃvartaka:

Ghana) | (Cubic) | khaṇḍukas | – | 12544 |

(Sūcī) | (Linear) | (rajjus) | – | 3136 |

(Pratara) | (Square) | (rajjus) | – | 784 |

(Ghana) | (Cubic) | (rajjus) | – | 196 |

Both together:

(Ghana) | (Cubic) | khaṇḍukas | – | 21952 |

(Sūcī) | (Linear) | (rajjus) | – | 5488 |

(Pratara) | (Square) | (rajjus) | – | 1372 |

(Ghana) | (Cubic) | (rajjus) | – | 343 |

Saṃvartita Loka-sthāpana (Figure of the Saṃvartita Loka)

*caturbhirguṇane tvāsāṃ, khaṇḍukānyekaviṃśatiḥ.*

sahasrāṇi navaśatī, dvipa–cāśatsamanvitā..

sahasrāṇi navaśatī, dvipa–cāśatsamanvitā..

- Lokaprakāśa, cantos XII, verse 141.