In the extracts below, population figures are highlighted in red. For comparison, the 2011 census counted 275 people in the parish of Meshaw.

The Parliamentary Gazetteer of England and Wales (1845) -

MESHAW, a parish in the hund. of Witheridge, union of South Molton, county of Devon; 5 miles south-east by south of South Molton. Living, a discharged rectory in the archd. of Barnstaple and dio. of Exeter; rated at £7 4s., returned at £139 18s. 9d. ; gross income £206. Patron, in 1835, the Rev. W. H. Karslake. Here is a daily school. Acres 2,680. Houses 36. A. P. £847. Pop., in 1801, 135; in 1831, 166. Poor rates, in 1838, £57 12s.

White's Devonshire Directory (1850) tells us -

"MESHAW is a small village, at the foot of a declivity, nearly 6 mile S. by S.E. of South Molton, and 14 miles N.W. by W. of Tiverton. Its parish increased its population from 166 in 1831, to 305 in 1841, and comprises 1770A. 3R. 30P. of land, part of which (about 700A.) was an open moor 20 years ago, but is now enclosed and cultivated. It is a hilly district, over which a new road was cut about ten years ago. The manor was held by the Avenells, in the 13th and 14th centuries, and was afterwards in moieties. The greater share now belongs to Richard Preston, Esq., and the rest belongs to John Brown, Wm. Adams, Robert Eastmond, and a few smaller freeholders. The Church (St. John Baptist,) was mostly rebuilt in 1838, except the tower, which has four bells. It is a small structure, and contains a handsome monument to James Courtenay, who died in 1683, at Meshaw House, now occupied by a farmer. The living is a discharged rectory, valued in K.B. at £7. 4s., and in 1831 at £206, in the patronage of the Rev. W. Karslake, and incumbency of the Rev. W.H. Karslake, B.A., who has a good residence, and 83A. 3R. 39P. of glebe. The tithes were commuted in 1838, for £110 per annum. In the village is a National School, and on the Moor is a school and chapel belonging to the Bible Christians, built in 1839."

A list of notable inhabitants in the 1850s can be found here. What caused the big increase in population by 1841? Navvies who stayed on after building the new road? People finding that the new road was good for business? The enclosure and cultivation of the moor?

Imperial Gazetteer of England and Wales 1870-1872 -

MESHAW, a parish, with a village, in South Molton district, Devon; 5 miles SE by S of South Molton, and 8 NE of Eggesford r. station. Post town, South Molton, North Devon. Acres, 1,751. Real property, £1,191. Pop., 250. Houses, 55.

The property is subdivided. The manor also is divided. Meshaw House, or Barton, was anciently the seat of the Courtenays, and is now a farm-house. The living is a rectory in the diocese of Exeter. Value, £197. Patron, the Rev. W. Karslake. The church was rebuilt in 1838; retains the tower of a previous edifice of 1691; consists of nave and chancel; and contains a memorial window to T. H. Karslake who fell at Sebastopol, and several monuments of the Courtenays. There are a chapel for Bible Christians and a national school.

Meshaw, a parish, with a village, in Devonshire, 5 miles. SE by S of South Molton, and 9 NE of Eggesford station on the L. & S.W.R. It has a post and money order office under South Molton; telegraph office, Whiteridge. Acreage, 2095; population of the civil parish, 169; of the ecclesiastical, 158. The manor is divided. Meshaw House, or Barton, was anciently the seat of the Courtenays, and is now a farmhouse. The living is a rectory in the diocese of Exeter; gross value, £220 with residence. The church was rebuilt in 1838, retains the tower of a previous edifice of 1691, consists of nave and chancel, and contains a memorial window to T. H. Karslake, who fell at Sebastopol, and a monument of the Courtenays. There is a Bible Christian chapel.