the size of the lattice.  If the lattice is weak, the column is simply deficient; so a formula for a hooped column is incorrect if it shows that the strength of the column varies with the section of the hoops, and, on this account, the common formula is incorrect.  The hoops might be ever so strong, beyond a certain limit, and yet not an iota would be added to the compressive strength of the column, for the concrete between the hoops might crush long before their full strength was brought into play.  Also, the hoops might be too far apart to be of much or any benefit, just as the lattice in a steel column might be too widely spaced.  There is no element of personal opinion in these matters.  They are simply incontrovertible facts.  The strength of a hooped column, disregarding for the time the longitudinal steel, is dependent on the fact that thin discs of concrete are capable of carrying much more load than shafts or cubes.  The hoops divide the column into thin discs, if they are closely spaced; widely spaced hoops do not effect this.  Thin joints of lime mortar are known to be many times stronger than the same mortar in cubes.  Why, in the many books on the subject of reinforced concrete, is there no mention of this simple principle?  Why do writers on this subject practically ignore the importance of toughness or tensile strength in columns?  The trouble seems to be in the tendency to interpret concrete in terms of steel.  Steel at failure in short blocks will begin to spread and flow, and a short column has nearly the same unit strength as a short block.  The action of concrete under compression is quite different, because of the weakness of concrete in tension.  The concrete spalls off or cracks apart and does not flow under compression, and the unit strength of a shaft of concrete under compression has little relation to that of a flat block.  Some years ago the writer pointed out that the weakness of cast-iron columns in compression is due to the lack of tensile strength or toughness in cast iron.  Compare 7,600 lb. per sq. in. as the base of a column formula for cast iron with 100,000 lb. per sq. in. as the compressive strength of short blocks of cast iron.  Then compare 750 lb. per sq. in., sometimes used in concrete columns, with 2,000 lb. per sq. in., the ultimate strength in blocks.  A material one-fiftieth as strong in compression and one-hundredth as strong in tension with a “safe” unit one-tenth as great!  The greater tensile strength of rich mixtures of concrete accounts fully for the greater showing in compression in tests of columns of such mixtures.  A few weeks ago, an investigator in this line remarked, in a discussion at a meeting of engineers, that “the failure of concrete in compression may in cases be due to lack of tensile strength.”  This remark was considered of sufficient novelty and importance by an engineering periodical to make a special news item of it.  This is a good illustration of the state of knowledge of the elementary principles in this branch of engineering.