(108, A.)

(109, A.)

For the scale for dwellings, we have, for those occurring between H and D -

r = 1/2(34 - t); (108, B.)

t = 24 - 2 r; (109, B.)

and for those between H and J. we have -

(108, C.)

(109, C.)

where, in each equation, r represents the riser, and t the tread, or net step.

By these formulae, the following tables have been computed:

### Stairs for Shops

 Rise. Tread. Ratio - Rise to Tread. 2. 24. 1 to 12. 3. 22. 1 ,, 7.33 3.50 2T. 1 ,, 6. 4. 20. 1 ,, 5. 4.50 19. 1 ,, 4.22 5. 18. 1 ,, 3.60 5.4 17.20 1 ,, 3.19 5.7 16.60 1 ,, 2.91 6. 16. 1 ,, 2.67 6.25 15.50 1 ,, 2.48 6.50 15. 1 ,, 2.31 6.70 14.60 1 ,, 2.18 6.90 14.20 1 ,, 2.06 7. 14. 1 ,, 2. 7.20 13.60 1 ,, 1.89
 Rise. Tread. Ratio - Rise to Tread. 7.40 13.20 1 to 1.78 7.6o 12.80 1 ,, 1.68 7.80 12.40 1 ,, 1.59 8. 12. 1 ,, 1.50 8.20 11.6 1 ,, 1.41 8.50 11. 1 ,, 1.29 8.80 10.40 1 ,, 1.18 9. 10. 1 ,, 1.11 9.30 9.40 1 ,, 1.01 9.60 8.80 1 ,, 0.92 10. 8. 1 ,, 0.80 10.50 7. 1 ,, 0.67 11. 6. 1 ,, 0.55 11.50 4.95 1 ,, 0.43 12. 3.58 1 1" 0.30

### Stairs for Dwellings

 Rise. Tread. Ratio - Rise to Tread. 2. 20. 1 to 10. 3. 18. 1 ,, 6. 3.50 17. 1 ,, 4.86 4. 16. 1 ,, 4. 4.50 15. 1 ,, 3.33 5. 14. 1 ,, 2.80 5.40 13.20 1 ,, 2.44 5.70 12.60 1 ,, 2.21 6. 12. 1 ,, 2. 6.25 11.50 1 ,, 1.84 6.50 11. 1 ,, 1 .69 6.75 I0.50 1 ,, 1.56 7. 10. 1 ,, 1.43 7.10 9.80 1 ,, 1.38 7.20 9.60 1 ,, 1.33 7.30 9.40 1 ,, 1.29
 Rise. Tread. Ratio - Rise to Tread. 7.40 9.20 1 to 1.24 7.50 9. 1 1.20 7.60 8.8o 1 ,, 1.16 7.70 8.6o 1 ,, 1.12 7.80 8.40 1 ,, 1.08 7.90 8.20 1 ,, 1.04 8. 8. 1 ,, 1. 8.10 7.80 1 ,, 0.96 8.30 7.40 1 ,, 0.89 8.50 7. 1 ,, 0.82 8.75 6.50 1 ,, 0.74 9. 6. 1 ,, 0.67 9.30 5.40 1 ,, 0.58 9.60 4.80 1 ,, 050 10. 3.90 1 ,, 0.39 10.50 2.20 1 ,, 0.21

These tables will be useful in determining questions involving the proportion between the rise and tread of a pitch-board.

For stairs in which the run is limited, to determine the number of risers which would give an easy ascent: Divide the run by the height, and find in the proper table, above, the ratio nearest to the quotient, and in a line with this ratio, in the second column to the left, will be found the corresponding riser. With this divide the rise in inches; the quotient, or the nearest whole number thereto, will be the required number of risers in the stairs.

Example. - For the stairs in a dwelling, let the rise be 12' 8", or 152 inches. Let the run between the extreme risers be 17' 2". To this, for the purpose of obtaining the correct angle of ascent, by having an equal number of risers and treads, add, for one more tread, say 10 inches, its probable width; thus making the total run 18 feet, or 216 inches. Thus we have for the run 216, and for the rise 152. Dividing the former by the latter gives 1 .42 nearly. In the table of stairs for dwellings, the ratio nearest to this is 1 .43, and in the line to the left, in the second column, is 7, the approximate size of riser appropriate to this case. Dividing the rise, 152 inches, by this 7, we have 21 5/7 as the quotient.

This is nearer to 22 than to 21; therefore, the number of risers required is 22.

When the number of risers is determined, then the rise divided by this number will give the height of each riser; thus, in the above case, the rise is 152 inches. This divided by 22 gives 6.909 inches for the height of the riser.

When the height of the riser is known, then, if the run is unlimited, the width of tread will be found in the proper table above. For example, if the riser is 7 inches or nearly that, then in the table of stairs for dwellings, in the next column to the right, and opposite 7 in the column of risers, is found 10, the approximate width of tread. By the use of equation (109, B.), the width may be had exactly according to the scale. For example, equation (109, B.) with 6.91 for the riser, becomes -

t = 24 - 2 x 6.91 = 10.18, or about 10 3/16 inches.

When the run is limited and the number of risers is known, then the width of tread is obtained by dividing the run by the number of treads. There are always of treads one less than there are of risers, in each flight.