College and Research Libraries


FRED HEINRITZ 

Rate of Growth 

For Library Collections 

A table is given to show conveniently up to a twenty-year interval 
and/ or a doubling of the collection the annual rate of growth of 
any library collection, given the growth period in years and the size 
of the collection at the beginning and end of this period. A formula 
for determining solutions beyond the limits of the table is also de-
rived and illustrated. 

THE BACKGROUND 

EVER SINCE RESEARCH LIBRARIES BEGAN 

in the United States, over three cen-
turies ago, their collections have on the 
average been doubling in size every six-
teen years. When these libraries were 
small, their librarians did not worry par-
ticularly about the rate of growth of 
their collections. But about the time of 
the Second World War, as building, and 
hence storage, costs increased, Rider and 
others sounded the alarm. 1 The problem 
has been exacerbated by the current 
squeeze on public funds. Academic and 
research librarians are by now tired of 
hearing, and are only too aware, that 
their collections are increasing at a geo-
metrical, rather than at an arithmetical, 
rate. 

Since an approximately constant (ex-
ponential) rate of growth represents 
the true state of affairs, it is of value to 
librarians planning building needs or 
engaging in other research involving 
comparisons and predictions of collec-
tion growth, to know the rate of growth 
for given situations. The accompanying 
table gives this result directly for peri-

Fred Heinritz is a professor at Southern 
Connecticut University. 

ods up to twenty years, saving the need 
for laborious computation. 

USE OF THE TABLE 

The only data needed to use the ta-
ble are the number of years covered 
( T), and the size of the collection at 
the beginning ( P) and end (A) of this 
period. As an example of use, suppose 
that a collection has grown from 
1,000,000 to 1,700,000 volumes over 
eight years. A/P = 1.700. The answer 
lies at the intersection of the 1.700 row 
and the 08 column. This value, 0.069, 
represents an annual growth rate of 6.9 
percent. 

ExTENSION OF THE TABLE 

For time periods exceeding twenty 
years and/ or A/ P values exceeding 
2.000, the rate of growth may be calcu-
lated by the following formula. (For 
the derivation, · see the Mathematical 
Note.) 

Annual Rate of Growth = 

Antilog [ log (;/P) ] - 1 

For example, if a library grows from 
136,260 to 87 4,999 volumes over a forty 
year period, the annual rate of growth 
is 4.76 percent. 2 The calculations are: 

I 95 



96 f C allege & Research Libraries • March 197 4 

TABLE 1 

ANNUAL RATE OF GROWTH OF A LIBRARY COLLECTION OVERT YEARS, 

BEGINNING WITH p AND ENDING WITII A ITEMS 

T 
A/P I 02 04 06 08 10 12 14 16 18 20 

--~-·--------------~-----------------------~--..-------------~-------·----... -----
l.Q2QI OaQZ.2 o.QlZ Q!QQ8 OaQOf! 
1.100 0.049 0.024 0.016 0.012 
1.150 0.072 0,036 0.024 0,018 
l.20Q Q.095 0,047 0.031 Q,Q23 
1.250 0.118 0.057 0.038 0.028 
1,300 o.140 0.068 0.045 0,033 
1.320 O.lbZ o. o:za O.Q!H o.o3a 
1.400 0.183 0,088 o.o58 0. 0'43 
1.450 0.204 0.097 0.064 0.048 
115QO 0,225 Q,1Q7 Q.070 OtQ5Z 
1.550 0.245 0.116 · 0.076 0,056 
1,600 0.265 0. 125 0.081 0.061 
l.QSO 0.282 0.13~ O.QIH 0.0~!2 
1,7001 0.304 0.142 0.092 0.069 
1,7501 0.323 0,150 0.098 0,072 
1 1 8QOI Q!342 0,158 - 0.103 0!076 
1,8501 0.360 0.166 0.10.8· 0,080 
1,9001 0,,378 0.174 0.113 0.084 
1.9501 0.396 0,182 0.118 0,087 
2,0001 0.414 0.189 0.122 0.091 

A. R. of G. 

= Antilog [log ( 87 4,9:~/136,260)] _ 1 

= Antilog [ log ( ~~215)] - 1 

= Antilog [ 
0

·
8
:

63
] - 1 

= Antilog [0'.02019] - 1 
= 1.0476 - 1 

= 0.0476 

MATHEMATICAL NoTE 

o.QQ5 Q.Q04 Q.003 o.oo3 0,003 . 0 1 0Q2 
0.010 0,008 0.001 0,006 0,005 0.005 
0.014 0,012 o.o1o 0,009 0,008 0.007 
0.018 0.015 o.o13 OtOll 0.010 0.009 

. 0.023 · 0.019 0.016 0,014 0,012 O.Oll 
0.027 0.022 0~019 0,017 0,015 0.013 
Q,Q30 0.025 o.o~z Q.Ql9 O.Ql7 o.ol5 
0.034 0,028 0.024 0,021 . 0,019 0.017 
0,038 0,0-31 0.027 0,023 0.021 0,019 
0.041 Q.Q34 o.Q~9 o.Q26 o.QZ3 Q I QZO · 
0,045 0,037 0.032 0,028 0,025 0.022 
0.048 0.040 0.034 .0. 030 0.026 0.024 
o.QSl Q.Q4l Q,Q36 o.on o.oza o.025 
0.054 0.045 0.039 0,034 0.030 0.027 
0.058 0.048 0.041 0,036 0~032 0.028 
0.061 0,050 0.043 0,037 0.033 0.030 
0.063 0,053 0.045 0,039 0,035 0,031 
0,066 0,055 0~047 0,041 0,036 0.033 
0.069 0,057 0.049 0,043 0,038 . 0,034 
0.072 0,059 0.051 0,044 0,039 0.035 

( 1 ) take the logarithm, ( 2) subtract log 
P, ( 3) divide by T, ( 4) take the anti-
logarithm, ( 5) subtract one. This gives 
us 

i = (A/P) 1fT - 1 

which for computational purposes is 
more conveniently expressed as 

. A fl [ log ( AlP) ] 1 1 = n 1 og T -

REFERENCES 

The mathematics for a constant rate 
of growth is identical to that for com-
pound interest. Thus the problem is, 
given A, P, and T, to solve 

1. Fremont Rider, The Scholar and the Future 
of the Research Library; A Problem and Its 
Solution (New York: Hadham Press, 1944). 

2. J. Periam Danton, Book Selection and Col-
lections; A Comparison of German and 
American University Libraries (New York: 

A=P(1+i)T 

for i. For both sides of the equation Columbia Univ. Pr., 1963), p.l03.