Do any scriptures clearly mention the year when Kaliyuga started?

Question

The popular belief of people is that Kaliyuga started around 3000 BC. What is the source such belief ? Which scripture tells about such date ?

Answer 1

In Āryabhaṭīya (3.10), Āryabhaṭa I states that -

ṣaṣṭyabdānām ṣaṣṭiryadā vyatītāstrayaśca yugapādāḥ /
tryadhikā viṁśatiḥabdāstadeha mama janmanp'tītāḥ //

When sixty times sixty years and three yugapādas (from the beginning of the yuga) had elapsed, twenty three years had then passed since my birth.

As per this śloka, 3600 yrs since the beginning of Kaliyuga had elapsed during the composition of Āryabhaṭīya. In their notes, KS Shukla & KV Sharma add,

This epoch corresponds to the mean noon at Ujjaiyinī, Sunday, March 21, 499 CE.
At this time Āryabhaṭa I was exactly 23 yrs of age. Āryabhaṭa I was therefore born on March 21, 476 CE.

Shukla & Sharma further add that as per the commentators of Āryabhaṭīya the year 3600 of Kaliyuga specifies the mean position of grahas computed from the parameters given in the Daśagītikāsūtra, didn't require any correction and that to the mean longitudes computed for a subsequent date a bīja correction was necessary. So, this epoch of 3600 Kali was the 'epoch of zero correction'. According to the commentators Sūryadeva, Parameśvara, and Nīlakaṇṭha, Kaliyuga year 3600 corresponds to Śaka 421 (i.e. 499 CE), it is the epoch of zero correction as well the time of composition of Āryabhaṭīya. As 499 - 3600 = -3101, Kaliyuga started at 3102 BCE, as per Āryabhaṭa I and commentators Sūryadeva, Nīlakaṇṭha, etc. The difference in the yr notation is because there's no 0 yr in Julian/Gregorian calendric system. On the other hand, when the yr is denoted on the integer scale, the prev. yr to +1 is 0, which is preceded by -1.
That's why although the yr is -3101 yet on English calendric system, it's 3102 BCE.

There's a popular online myth as per which Sūryasiddhānta states that Kaliyuga began at February 18, 3102 BCE. But Sūryasiddhānta doesn't stated anything like this. The people who claim that Sūryasiddhānta mentions this date, never quote the exact ślokas from the śāstra.

In Śiṣyadhīvṛddhidatantra (1.12-14), Lalla mentions Kaliyuga began 3179 yrs before start of Śaka era, while explaining how to calculate the Kali-ahargaṇa -

navādricandrānalasaṁyuto bhavecchakakṣitīśābdagaṇo gataḥ kaleḥ /
divākaraghno gatamāsasaṁyutaḥ khavahrinighnasthitibhiḥ samanvitaḥ //
pṛthak kṛta saṅguṇito'dhimāsakaistato vibhakto divasaiḥ sahasragoḥ /
phalādhimāsaiḥ sahitau dinīkṛtaiḥ punaḥ pṛthakastho guṇitastithikṣayaiḥ //
niśākarāhaiḥ sa hyataḥ phalava mai biṁvarjito'rkābhyodayādahargaṇaḥ /
bhavatyayaṁ bhārgavavārapūrvakaḥ sadaiva laṅkāviṣaye kalergataḥ //

The Śaka year added to 3179 gives the solar years elapsed since the beginning of the Kaliyuga. Multiply the no. by 12 and add the months (beginning from Caitra) passed in the current yr. Multiply the sum by 30 & add the no. of tithis elapsed (since the last amāvasay). Put down the result at two places. (At one place), multiply it by the no. of intercalary months (in a yuga) and divide by the no. of solar days. The result is corresponding no. of intercalary months. Change it into days & add to the result at the 2nd place. Put down the sum at two places. (At one place), multiply it by the no. of omitted tithis (in a yuga) & divide by the no. of tithis (in the same period). The result is corresponding no. of omitted tithis. Subtract it from the sum at the 2nd place. The remainder is always the number of civil days or ahargaṇa elapsed since the beginning of Kaliyuga (till given day) & is counted at Friday at (mean) sunrise at Laṅkā.

The Śaka era is known to have begun at 78 CE. (78 - 3179 = -3101)
So Kaliyuga started at 3102 BCE, as per Lalla.

Bhāskara II also states that Kaliyuga began at -3179 Śaka, in Siddhāntaśiromaṇi (1.28)

yātāḥ ṣaṇmanavo yugāni bhamitānyanyadyugāṅghritrayaṁ
nandādrīnduguṇāstathā śakanṛpasyānte kalervatsarāḥ /
godrīndvadrikṛtāṅkadasranagagocandrāḥ 1972947179 śakābdānvitāḥ /
sarve saṅkalitāḥ pitāmahadine syurvarttamāne gatāḥ //

Here, Bhāskara II says that before start of Śaka era, 6 Manus, 27 mahāyugas, 3 yugas (Kṛta, Tretā, & Dvāpara), and 3179 yrs of the 4th yuga (Kali) have elapsed, totalled 1972947179 years.

Just like Lalla, Bhāskara I mentions the method to calculate Kali-ahargaṇa in his śāstra Laghubhāskarīya, where he specifies that Kali started at -3179 Śaka.

navādrayokāgnisaṁyuktā śakābdā dvādaśāhatā /
caitrādimāsasaṁyuktāḥ pṛthagguṇyā yugādhikaiḥ //
te ca ṣaṭtrikarāmāgninavabhūtendavo yuge /
bhāgahāro'bdhivasvekaśarāḥ syurayutāhatāḥ //
adhimāsānpṛthakstheṣu prakṣipya triṁśatāhate /
yuktvā dināni yātāni pratirāśya yugāvamaiḥ //
saṅguṇyyāmbarāṣṭeṣudvyaṣṭaśūnyaśarāśvibhiḥ /
chedaḥ khāṣṭaviyadvyomakhakhāgnikharasendavaḥ //
labdhānyavamarātrāṇi teṣu śuddheṣvahargaṇaḥ /
vāraḥ saptahyate śeṣe śukādirbhāskarodayāt //
~ Laghubhāskarīya (1.4-8)

Add 3179 to the (no. of elapsed) yrs of the Śaka era, (then) multiply (the resulting sum) by 12, and (then) add the (number of lunar) months (expired) since the commencement of Caitra. Set down (the result thus obtained) at (two) separate places; multiply (one) by (the no. of) intercalary months in a yuga, which are 1593336 in a yuga ; add the (resulting complete) intercalary months to result placed at the other place. Then multiply (that sum) by 30 & (to the product) add the tithis expired (of the current month). Set down (the result thus obtained) in two places ; multiply (one) by the (no. of) ommitted tithis in a yuga, i.e., by 25082580, and divide by 1603000080. The resulting (complete) omitted lunar days when subtracted from the result put at the other place give the (required) ahargaṇa. The remainder obtained on dividing (the ahargaṇa) by 7 gives the day beginning with Friday at sunrise (at Laṅkā).

With jyotiṣācāryas such as Āryabhaṭa I, Bhāskara I, Bhāskara II, Lalla, Nīlakaṇṭha Somayājī, Sūryadeva, etc. concurring on the fact that Kaliyuga began at 3102 BCE, there is no point in entertaining any doubt. It is possible for one jyotiṣācārya to be wrong, but the entire jyotiṣaparamparā is concurring on this fact. Therefore, this info is undoubtedly valid.

In Āryabhaṭīya (3.11), Āryabhaṭa I states that the yuga begins at the Caitra Śuklapratipāda

yugavarṣamāsadivasāḥ samaṁ pravṛttāḥ tu caitraśuklādeḥ /
kālaḥ ayam anādyantaḥ grahabhaiḥ anumīyate kṣetre //

The yuga, the year, the month, and the day commenced simultaneously at the beginning of the śuklapakṣa of Caitra. This time, which is without beginning and end, is measured with the help of grahas and the nakṣatras on the celestial sphere.

Hence, Kaliyuga began at Caitra Śuklapratipāda in 3102 BCE, at mean sunrise (at Laṅkā).
In Śaka, 3102 BCE (-3101) is (-3101 - 78 = -3179) ⇒ -3179 Śaka
In Vikrama, 3102 BCE (-3101) is (-3101 + 57 = -3044) ⇒ -3044 Vikrama

Note: Laṅkā (not Sri Lanka) is located at the intersection of equator & meridian through Ujjaiyinī. Refer Āryabhaṭīya (4.13-14) & its comm. for further info. on that.